In 1416, Kim In-u was appointed Inspector of the Mu-leungdo region and sent to Mu-leungdo (Ulleungdo) to bring back the people living there. Koreans feared that settlers on the island would invite raiding Japanese pirates. In 1417, Kim In-u returned from his inspection trip, but the record says that he returned from Usando, not Mu-leungdo, suggesting that "Usando" was the name then being used for Ulleungdo.
Here is the 1417 passage from the Records of King Taejong:
Notice that the passage says that Kim In-u returned from Usando, not Mu-leungdo, where he was ordered to go by King Taejong. This suggests that the residents of Ulleungdo were calling the island "Usando," not "Mu-leungdo." This would explain why twelve people who came by boat to the mainland in 1412 said they were from Yusanguk-do, which was most likely a misspelling of Usanguk-do. They also said they had grown up on a neighboring island called "Mu-leungdo." You can see a post on that incident here.
Source: 「太宗實錄」卷 三十三, 太宗 十七年 二月 壬戌條
February 5, 1417
按撫使金麟雨還自于山島, 獻土産大竹、水牛皮、生苧、綿子、檢樸木等物, 且率居人三名以來。 其島戶凡十五口, 男女幷八十六。 麟雨之往還也, 再逢颶風, 僅得其生
안무사(按撫使) 김인우(金麟雨)가 우산도(于山島)에서 돌아와 토산물(土産物)인 대죽(大竹)·수우피(水牛皮)·생저(生苧)·면자(綿子)·검박목(檢樸木) 등을 바쳤다. 또 그곳의 거주민 3명을 거느리고 왔는데, 그 섬의 호수[戶]는 15구(口)요, 남녀를 합치면 86명이었다. 김인우가 갔다가 돌아올 때에, 두 번이나 태풍(颱風)을 만나서 겨우 살아날 수 있었다고 했다.
Inspector Kim In-u returned from Usando and brought local products as tribute, including large bamboo, seal skins, raw ramie cloth, silk wool, and 검박목. He also brought back three residents of the place. There were fifteen families living on the island for a total of 86 men and women. On his way back from the island, Kim In-u ran into two typhoons and barely made it back alive.
In previous records, Ulleungdo had been referred to by various names, including Usan-guk, which means the "Country of Usan," but the above passage was the first time that Ulleungdo had been referred to as Usando, which means the "Island of Usan." This is significant because it is saying Usan is an island, not a country that could possibly included some rocky islets ninety-two kilometers away. Actually, in 1412, there was already a suggestion that "Usan" was an island, not a country, when residents of Ulleungdo referred to their island as "Yusanguk-do," which means the "Island of Yusanguk."
Koreans claim that Usando is Dokdo/Takeshima, but Dokdo/Takeshima is just a group of rocks with no soil to grow any of the products that the record says that Kim In-u brought back from the island. Koreans have tried to explain away the discrepancy by saying that Usan refers to a country, not an island, but the above record clearly says that Usan was an island, not a country. Even the Samguksagi said that Usan-guk was just another name for Ulleungdo. You can see another post on the subject here.
By 1417, records show that Ulleungdo was made up of two islands, one main island and a smaller neighboring island. Though there seems to be some confusion about the names of the islands, records in 1412 and 1417 suggest that the name of the main island was Usando, that its neighboring island was Mu-leungdo, and that both islands had settlers on them at one time or another. In future records, these names will be used together to refer to the Ulleungdo island group, and Usando will often be listed first, suggesting that it is the larger of the two islands. So far, there has been nothing in the records to suggest that either Usando or Mu-leungdo is a referrence to Dokdo/Takeshima.