Since my movie review for Noryang: Deadly Sea is public, whereas my additional commentary and trivia article is an early access for my tipper-tier patrons, I've prepared this article is for my supporter-tier patrons, and it will remain permanently patron-exclusive.
Admiral Yi Sun-sin's exploits and contribution to Imjin War are way overblown in popular discourse, so much so that he is often lauded as single-handedly stopping the Japanese dead in their tracks. To have a correct big picture view of Imjin War, it is crucial to understand the roles Yi Sun-sin played in the war, neither downplaying nor overhyping his contribution.
But first, a timeline of notable Yi Sun-sin's activities during the war:

May 23, 1592: Japanese invasion fleet crossed Korea Strait and laid siege to Busan. Beginning of Imjin War.
May 25, 1592: Yi Sun-sin was alerted by report from Won Gyun (원균 or 元均) about significant Japanese naval activities near the coasts of Korea and realised that something had gone awry. He heightened the preparedness level of all personnel under his command, while awaiting formal orders from the Joseon court.
May 30, 1592: News about the fall of Busan and Dongnae reached Yi Sun-sin.
June 10, 1592: Yi Sun-sin mustered his fleet at Gyeongsang Jwasuyeong (his naval base) in preparation for the possible Japanese incursion into Jeolla Province. However, he changed his mind after receiving a call for help from Won Gyun, and set off his fleet to rendezvous with Won Gyun's remnant fleet at Geojje Island.
June 16-17, 1592: Battle of Okpo (which was actually three consecutive skirmishes: Okpo, Happo, and Jeokjinpo). Yi Sun-sin won his first naval victory against the Japanese.
July 8, 1592: Battle of Sacheon. First recorded use of turtle ships.
July 10, 1592: Battle of Dangpo. Tokui Michiyuki (得居通幸), older brother of Japanese commander Kurushima Michifusa (来島通総), was believed to be killed during this battle.
July 13, 1592: First Battle of Danghangpo. Yi Sun-sin was able to determine the whereabouts of the remnants from Battle of Dangpo thanks to the tips from several Japanese defectors.
July 15, 1592: Battle of Yulpo. Yi Sun-sin pursued and thoroughly defeated a Japanese fleet. Commander of said fleet, Shikoku Shima-no-Kami (四国志摩守), fled to a nearby island and committed suicide. Japanese invasion fleet reduced its activities afterwards, so Yi Sun-sin returned to Jeolla Province.
July 22, 1592: Yi Sun-sin's threat to the Japanese had become serious enough to warrant several notable Japanese naval commanders being summoned to Busan to discuss the next course of action. After the military council, Japanese invasion fleet became more active around the coasts of Gyeongsang Province once again.
August 12, 1592: Yi Sun-sin received intelligence about new Japanese naval activities near Gadeok Island and Geoje Island. He set sail to Noryangjin to join force with Joseon fleets under Won Gyun and Yi Eokgi (이억기 or 李億祺).
August 13, 1592: The combined Joseon fleet arrived at Gyeonnaelyang and detected its Japanese counterpart led by Wakisaka Yasuharu (脇坂安治).
August 14, 1592: Battle of Hansan Island. This battle was the first (and only) serious Japanese attempt to defeat Joseon navy during the first invasion, and it ended in an utter disaster.
August 16, 1592: Battle of Angolpo. The combined Joseon fleet defeated the fleets of Katō Yoshiaki (加藤嘉明) and Kuki Yoshitaka (九鬼嘉隆) coming to reinforce Wakisaka Yasuharu.
August 20, 1592: Toyotomi Hideyoshi (豊臣秀吉) received news about Wakisaka Yasuharu's defeat. He ordered Wakisaka Yasuharu to cease all offensive activities and focus on defending of Busan Japanese Fortress, and dispatched Tōdō Takatora (藤堂高虎) to reinforce the invasion fleet.
On the other hand, as the west of Gadeok Island was finally cleared of Japanese ships, Yi Sun-sin set his sight on his next objective—Japanese headquarters at Busan, and began preparation.
October 5, 1592: First Battle of Busan (which was actually five skirmishes and one major attack: Jangnimpo, Hwajungumi, Dadaepo, Seopyeongpo, Jeolyeong Island, and finally Busan itself). Combined Joseon fleet destroyed more than a hundred vacated Japanese ships as Japanese troops refused to engage the Koreans in naval battle. However, Joseon fleet was unable to make landing or seriously threaten Busan Japanese Fortress.
After the battle, Yi Sun-sin ceased activities to wait for the arrival of Ming army.
March 1, 1593: (Erroneously) believing a final Japanese defeat was imminent, Joseon court ordered Yi Sun-sin to cut off the line of retreat of the Japanese invasion force to prevent them from returning to Japan. Yi Sun-sin thus set his sight on the primary Japanese naval base at Ungcheonpo, which was garrisoned by Wakisaka Yasuharu, Katō Yoshiakiand and Kuki Yoshitaka.
March 9, 1593: Battle of Ungpo/First attack on Ungcheon. Yi Sun-sin once again joined force with Won Gyun and Yi Eokgi at Gyeonnaelyang Strait, and set off to attack Ungcheonpo. However, Japanese fleet refused to engage and hid deep inside the bay, ignoring Yi Sun-sin's attempts to lure them out into the open. With no one to fight, combined Joseon fleet retreated without achieving anything.
March 15, 1593: Second attack on Ungcheon. Japanese still refused to engage the Koreans.
March 20, 1593: Third attack on Ungcheon. A few Japanese ships gave in to the provocations and came out to meet Joseon fleet in battle—and suffered the consequences. After being peppered by arrows and suffered heavy casualties, they broke free of Joseon encirclement and fled.
March 24, 1593: Fourth attack on Ungcheon. Yi Sun-sin sent two warships to threaten an amphibious assault as a diversionary feint. Unfortunately, his diversionary forces ignored order and actually made landing. They were immediately surrounded by the Japanese on land and suffered heavy casualties. Yi Sun-sin's nephew Yi Bun (이분 or 李芬) also separately reported that one of the ships capsized after they collided into each other.
Despite the blunder, outwardly Yi Sun-sin still reported today's battle as a victory. He only secretly vented frustrations in his personal diary.
March 30, 1593: Fifth attack on Ungcheon. Japanese continued to refuse engagement. Yi Sun-sin found out Won Gyun's attempt to massacre Korean civilians to pass them off as Japanese soldiers, leading to a feud between the two.
April 7, 1593: Sixth attack on Ungcheon. Yi Sun-sin claimed that the Japanese completely abandoned their ports and ships and retreated to a more defensible high ground. However many of them were killed after being subjected to a naval bombardment.

April 23, 1594: Second Battle of Danghangpo. Joseon fleet destroyed 31 anchored Japanese ships. Yi Sun-sin was criticised for violating the ceasefire.
November 1594: The last of the Ming army pulled out of Korea. Immediately after the Chinese left, Yun Du-su (윤두수 or 尹斗寿) proposed a plan to evict the Japanese from the lightly defended Jangmunpo and Yeongdeungpo naval bases on Geoje Island, as the Koreans were against peace talks and wanted to finish the war with their own hands.
Gwon Yul (권율 or 權慄), Kim Deok-ryeong (김덕령 or 金德龄) and Yi Sun-sin were appointed as the commanders of the new offensive.
November 11, 1594: Battle of Jangmunpo/First attack on Jangmunpo and Yeongdeungpo. Joseon fleet destroyed two Japanese warships, causing others to abandon ships and flee to the island. Joseon fleet then proceeded to destroy the abandoned ships as well.
November 12, 1594: Second attack on Jangmunpo and Yeongdeungpo. Both sides withdrew after a standoff without actually doing battle. However, after Joseon fleet withdrew, Japanese fleet launched a surprise attack with two small ships and successfully burnt down an anchored Joseon warship.
November 13, 1594: Third attack on Jangmunpo and Yeongdeungpo. Combined Joseon fleet engaged in a ship-to-shore battle with the Japanese, which was fierce but ultimately inconclusive.
November 14, 1594: Fourth attack on Jangmunpo and Yeongdeungpo. Combined Joseon fleet launched an amphibious assault against Japanese positions in concert with Joseon army. However, they were unable to overcome Japanese fortifications and had to retreat.
November 16, 1594: Skirmish near Jangmunpo and Yeongdeungpo. Four Joseon ships encountered two Japanese ships during a scouting mission. Japanese troops immediately abandoned the ships and swam to the island. The abandoned ships were torched by Joseon ships.
November 18, 1594: Yi Sun-sin returned to Hansan Island. The failure of this military campaign eventually caused Yun Du-su to be sacked.
Yi Sun-sin's lack of deliverable results during this period was also framed by his political enemies in Joseon court as him getting arrogant and indolent after initial victories, which caused Joseon king to grow increasingly impatient with him.

January 29, 1597: Second Battle of Busan. Yi Sun-sin reportedly set fire to some parts of Busan Japanese Fortress, killing 34 Japanese troops and burning down thousands of buildings, several gunpowder depots, as well as uncountable amount of supply and other materiel. However, the reliability of his report was immediately cast into doubt by his political enemies.
March 8, 1597: After Joseon court was tipped off by Konishi Yukinaga (小西行長), Gwon Yul ordered Yi Sun-sin to ambush Katō Kiyomasa (加藤清正) during his transit to Korea, and he had no choice but to comply after trying to ignore the order several times.
Konishi Yukinaga's tip was actually genuine (it wasn't a conspiracy to get rid of Yi Sun-sin). However, due to various delays Katō Kiyomasa already arrived in Korea one week before Yi Sun-sin receiving his order.
April 27, 1597: Yi Sun-sin was blamed for failing the mission and imprisoned.
May 16, 1597: Yi Sun-sin was pardoned and rejoined the military as a rank-and-file soldier.
August 27, 1597: Battle of Chilcheollyang. Joseon navy suffered a catastrophic defeat.
September 13, 1597: Yi Sun-sin was reinstated as the admiral of Joseon navy.
October 26, 1597: Battle of Myeongnyang. The high point of Yi Sun-sin's military career, where he defeated a huge Japanese fleet with only twelve or thirteen ships. Kurushima Michifusa perished in this battle.
October 31, 1597: Yi Sun-sin retreated into Yellow Sea to evade the still-stronger Japanese fleet (as well as Japanese army intruding into Jeolla Province), arriving at Gogunsangundo Islands on this day.
March 28, 1598: Yi Sun-sin returned to the southern coasts of Korea. He set up a new naval base at Gogeumdo Island and began to rebuild his fleet.
July 14, 1598: Ming navy led by admiral Chen Lin (陳璘) arrived at Seoul.
August 19, 1598: Ming navy arrived at Gogeumdo Island.
August 20, 1598: Battle of Jeolido Island. Yi Sun-sin defeated a Japanese fleet in a meeting engagement near Jeolido Island and beheaded 71 Japanese troops. However, Ming navy refused to help, citing bad weather as an excuse. Chen Lin was utterly enraged by the indolence of his own fleet, to the point that Yi Sun-sin had to gift some Japanese heads to the Ming admiral to pacify him (Yi Sun-sin complained about this in a report to Joseon court).
October 18, 1598: Siege of Suncheon began.
November 7, 1598: Siege of Suncheon failed. Both Ming army and allied navy retreated.
December 5, 1598: Chen Lin informed Yi Sun-sin that Konishi Yukinaga's was about to return to Japan, and urged him to prevent his retreat.
December 6, 1598: Allied fleet returned to Suncheon Fortress and set up a naval blockade.
December 16, 1598: Battle of Noryang.
A holistic overview of Yi Sun-sin's activities during Imjin War can easily unveil and debunk many myths surrounding the legendary admiral, including but not limited to:
Yi Sun-sin never actually cut off the Japanese naval supply line, since he was mostly active at the west side of said supply line. The only time Yi Sun-sin could plausibly threaten Japanese supply line was during the First Battle of Busan. However, cutting off a supply line requires a long term blockade, so a battle that lasted only a single day simply wouldn't cut it.
The claim that Yi Sun-sin remained undefeated is greatly exaggerated, as Yi Sun-sin certainly had his fair share of losing battles. His most obvious (but nevertheless overlooked) defeat was the failed Siege of Suncheon, although Battle of Busan, Battle of Ungpo, and Battle of Jangmunpo can also be counted among his failures (Battle of Busan is often spun into a Korean victory, whereas the other two battles are conveniently left unmentioned).
The claim that Yi Sun-sin "never lost a single vessel" is also exaggerated, as he did lose a ship to Japanese arson attack during Battle of Jangmunpo, to say nothing about ship lost during the famous Battle of Noryang.
Moreover (although this can't be easily discerned from studying Yi Sun-sin alone), there are some popular level history books such as Samuel Hawley's The Imjin War that tried to frame Yi Sun-sin's naval campaigns as a contest for the Yellow Sea. In other words, Toyotomi Hideyoshi wanted to establish a second naval supply line from Busan to Pyongyang through Yellow Sea to support the offensive on land, but Yi Sun-sin's naval victories ruined that plan. Thus, without this Yellow Sea supply line, Japanese army was forced to rely on inconvenient land supply line that faced constant harassment from Korean guerrillas and, as a result, quickly ran into serious supply shortage, setting the stage for its eventual downfall.

This framing thus elevated the effect of Yi Sun-sin's naval victories from safeguarding Korea's coasts into altering the strategic course of the entire war. Unfortunately, there is one big hole to this narrative: It simply assumes, without evidence, the existence of such a plan because it seems to make sense, without actually checking with Japanese sources to see if the plan to establish Yellow Sea supply line really existed (it didn't).
In truth, Yi Sun-sin's contributions to Imjin War were mostly localised. For the first invasion, the most direct effect of his naval victories was that he prevented the Japanese from pillaging the southern coasts of Korea, saving countless innocent lives. In addition, by October 1592 (after Battle of Busan) he had essentially forced the Japanese navy to give up all activities west of Gadeok Island, greatly reducing its available area of operations. This in turn ensured that Japanese invaders would never be able to threaten Jeolla Province from the sea.
For the second invasion, Yi Sun-sin's miraculous victory at the Battle of Myeongnyang stroke fear into the Japanese, for although Yi Sun-sin immediately retreated into the Yellow Sea with the Japanese fleet on his tail, never once did the Japanese pursuers seriously attempt to attack his much smaller fleet. In addition, while Yi Sun-sin did not completely halt the Japanese advance, nor did the damage inflicted on the Japanese fleet crippling, he did greatly slow them down and bought enough time to rebuild Joseon fleet and await Ming reinforcement.