They both are. Sony owns rights to Bloodborne and Nintendo to Bayonetta 2.
I usually call a game the first party game, when it is developed internally, I assume there can be other interpretations, I never really cared to check. Also Nintendo surely doesn't own Bayo IP, does it?
Owning rights doesn't say a thing, but who developed a game, tells a story.
You wouldn't buy Byo developed by Bethesda for example or would you?
edited:
what I'm trying to say, it's hiding behind words, we all now they aren't first party games and could have been made in other ways, they are only funded for exclusivity, but we insist that they are because?