This is the list, you can put Metroid where you like..

In House:
- EAD1
- EAD2
- EAD3
- EAD4
- EAD5
- EAD Tokyo
- SPD PG 1
- SPD PG 2
- SPD PG 3
- SPD PG 4

First Party

- Intelligent Systems
- Retro Studios
- Brownie Brown
- Monolith Soft
- Nintendo Software Technology (NST)
- Project Sora

Second Party or exclusive/licensed parties (It really depends and at one time or another they may not be but here's a list of common Nintendo second parties)

- Alpha Dream
- Ambrella
- Camelot
- Creatures Inc.
- Game Freak
- Genius Sonority
- HAL Laboratory
- Monster Games
- NDCUBE
- Noise
- Next Level Games
- Skip
- Sora
- Treasure
- Team Ninja/Project M

Game Freak itsn't first party?

if 1st party is developed published and licensed by nintendo and 3rd party is only licensed by nintendo. wouldnt that make published and licensed by nintento 2nd party or something?

You can't simply nail it as first or third party - it's both, and should be described as such (yhird party developed, first party published).

misterd, that's also wrong, it's a Project M game which means Team Ninja/SPD PG1 title which is a licenced second party.

It must be a second party game. It can't be third party.

publisher wise
1st party - the console manufacturer
3rd party - not the console manufacturer

developer wise
1st party - a subsidiary of the console manufacturer and one that the manufacturer has controlling interest
2nd party - exclusive to a console via contractual agreement
3rd party - not tied to any system

Metroid: Other M is first party published and since they named the development collaboration Project M it should be considered second party developed.

Well, as long as it's published by Nintendo, it's first party, just like F-Zero GX.

This game is being made by Nintendo... Team Ninja is helping them. Nintendo clearly is in charge with this game. Read the interviews.

Game Freak have made games for other systems.

Even post-Pokemon, but I guess the Game Freak-Nintendo relationship is tight.

Chrizum is correct.

It's not different from the N64 days with Nintendo having (IIRC) a 49% stake in Rare.  All of Rare's games were 2nd party games.