I feel I should talk this out with those who may know better than me, because this would affect my vote greatly; were Rare's games on SNES and N64 first or second party?
I feel like the answer is pretty complicated.
They would technically be considered a second party developer, but Nintendo also owned 49% of them. That is as close as you can get to 1st party without being first party. If you exclude Battletoads and the arcade version of Killer Instinct, they exclusively made games for Nintendo consoles from 1994 to 2002. Developers like HAL Laboratory would also be second party.
The more I look at it, it doesn't make a lot of sense to me to exclude second party games published by Nintendo. You would have to exclude the Donkey Kong Country games, Earthbound, all the Kirby games, Super Mario RPG, all the Fire Emblem games, and others on SNES. On N64, there are the Banjo games, Blast Corps, Donkey Kong 64, Goldeneye, Kirby 64, Mario Golf, all the Mario Party games, Mario Tennis, Paper Mario, Pokemon Snap, Sin and Punishment, and Smash Bros, among others. I didn't even look at any of the other consoles, but I bet it is the same there.
For me, I think it would just be easiest and best to just look at everything published by Nintendo for each console.