I literally just finished Bioshock yesterday. 

What puzzle solving did it have? You had to do certain tasks, but they weren't puzzles, they were going to certaing goals while shooting along the way. I'm not saying it's bad, it's just not what Metroid is. 

Exploring yes, but still more limited than Metroid. Metroid games are laybrinths that you navigate and slowly unlock and explore. Bioshock was more of levels that you complete certain tasks within, by shooting.

In Metroid Prime 3 you shoot enemies to progress through the level, you use a first person view and you can see your gun. Sounds like a FPS to me, having more focus in adventuring than other FPSs on the market doesn't change that fact.

Bioshock still had the hacking puzzles, plenty of exploration, and things like using the fire power to melt the iced-over door (similar to using a new beam to open a door in Metroid) and finding the parts for the bomb, (and, like Prime 3) etc.It wasn't as liberal as Prime 3 with these elements, but it was more than just shooting.

Heck, in Metroid you shoot just to open doors, and in most rooms you shoot enemies too.

But when you compare a game like Metroid to Killzone or Halo or Resistance, the games really have nothing in common. Sure they both have weapons that fire projectiles, but really that's where the comparisons end.

As for Bioshock, it's kind of the exception, but I will still address it. The puzzles in that game, or using of powers was optional. You could advance throught that game really without using your powers to melt Ice, or doing the hacking puzzle.

Metroid pretty much requires that you do the puzzles.

Many games within the same genre have little in common besides certain core traits. Monster Hunter and Demon's Souls are both ARPGs, yet they are very different.

And things like melting the first ice door and buildings the bomb in Bioshock weren't optional.

It's not like you can compare Killzone, Halo or Resistance to Wolfenstein 3D(which was the first FPS, therefore, the model for games in this genre) either. If that's what defines a FPS, then no current FPS is part of the FPS genre. Also, in Bioshock you may be able to advance through the game without using your powers, but it's not like in Metroid you can complete the game without shooting enemies, so that kind of argument goes both ways.

The truth is that the term FPS means First Person Shooter. You play Metroid in a first person view. You shoot enemies. Therefore, it's a FPS.

single player: Halo: Reach
multiplayer: Halo 4

- I voted Halo 4 overall because I like its single player more than Reach's multiplayer

Halo3 took me from being totally meh about Halo to being gay for it.