Fish Classifications

Non-white-rarity fish in the Fishing Log have their their own classifications. Not all fish of the same rarity have the same classification.

The English names are a bit of an educated guess, but reasonable. The EN fish tooltips and bonuses/bonus descriptions are a bit superfluous but the JP ones are rather explicit with how they’re named, explained more below.
The EN names for ocean fishing classifications is not official but rather is based on the names of the EN and JP bonuses. One of them is pretty simple and obvious, while the other is much more of an educated guess.

Classifications:
ヌシ (nushi) – big fish
オオヌシ (ōnushi) – legendary fish
珍魚 (chin-gyo) – rare fish (ocean fishing green fish)
伝説魚 (densetsu-gyo) – fabled fish (blue fish)

The bonus for catching a rare fish is just called ‘珍魚’ in JP, literally the name of the classification. In EN this bonus is called ‘A Rare Catch’ so it has the term in it with the usual flavour that EN translation adds.
The name for fabled fish in JP translates to ‘legendary fish’ but since the official EN term for ōnushi is ‘legendary fish’, ‘fabled fish’ is the next best guess due to it being a word in one of the two bonuses about them.
The bonus for catching one of them is called ‘伝説魚’ in JP, which is just ‘legendary fish’ and we can’t use that. In EN this one is called ‘Catch of a Lifetime’ which doesn’t provide anything useful..
The bonus for three of them being caught among your party is called ‘伝説の大漁旗’ in JP and also uses the ‘legendary’ (伝説) term. In EN this one is called ‘Fabled Fishers’ so we take the ‘fabled’ from its EN counterpart.

Ultimately, however, this is just a guess and could be completely wrong. These are often just called ‘blue fish’ among players anyway, but to people not aware that there are fish that are blue-rarity (since it’s just seven of the rarest fish in the game) using that term comes with its own potential confusion.
I think it’s quite possible this is not the official term, but I don’t have a better or more reasonable one, at least one that won’t cause confusion. ‘Legendary fish’ would obviously work, but we got confirmation of ōnushi officially being called ‘legendary fish’ when Fish Eyes got changed in patch 5.4. Given that we’ve had legendaries since 2.4, who knows how long it will take to get something clearly saying the official name of fabled fish.

A large part of why this type of confusion is nonexistent with JP terms is that the JP fish have tooltips saying at the end what classification they are. For the ocean fishing ones this classification the tooltip directly states is the same as the name for the bonuses.

Examples:
The Horned King: 「砂の川」のヌシ (nushi of the River of Sand)
Cinder Surprise: 「砂の川」のオオヌシ (ōnushi of the River of Sand)
Cieldalaes Geode: シェルダレー諸島沖合の稀少魚 (chin-gyo from the Cieldalaes)
Elasmosaurus: メルトール海峡北の伝説魚 (densetsu-gyo from the Northern Strait of Merlthor)

The end of the JP tooltips have a line that is ’[classification] of [fishing spot]’ for every fish that is one of the four classifications above in the Fishing Log.

The EN tooltips for big fish having things like “lord/lady/king/queen of [location]” as part of the flavour text in the tooltip, and this appears to be the localization of that, but it is not nearly as clear or consistent.