I thought the Big one was called generally as "Sea Serpent", and just like some have mentioned, have never been called as Leviathan in the game.
I think of it (From what I've seen/heard in the game) as some Catastrophic being, some kind of natural disaster. Maybe like Nidhogg in Ishgard.
I hope that the both of them (Well, Leviathan surely will) will have a much more appearance and place in the game, maybe make him some kind of the ruler of the sea for underwater content. Whatever SE can come up with that makes sense.