These three are really what make a Nintendo "First party game". Anything that's an EPD production is generally first party, which is 95% of Nintendo published games, whether that be internal creations or out-house/contracted productions.
- A game at least produced by Nintendo EPD, and we are including Astral Chain, Pokémon, and Buddy Mission BOND;
- A game at least (co-)designed by Nintendo EPD, and we are including Pikmin 4 and Metroid Dread;
- A game designed and primarily developed by Nintendo EPD, restricting the definition to mainline Zelda, Mario and Splatoon,
This topic settled with the second definition, which is probably the most sensible in a business sense, though the two EPD-centric ones define a "Nintendo game" in a more creative sense.
It all goes back to Nintendo fans not really understanding the true extent of EPD's operations. It's not just Mario and Zelda games. It's a global development operation that has hundreds of internal development staff, departments for IT, Coordination, Sound, Art, and Administration. Ties and oversight to doezens of subsidiary and partner studios (MonolithSoft, Retro, etc.) and is involved with nearly every game Nintendo publishes worldwide.
EPD is all Nintendo first party.