What defines a FPS game is the view. If you're looking through the player's eyes, while holding a weapon of some sort, it is an FPS game, which would make MP games all FPSs. But some people say that there are other certain requirements for an FPS. For example, MP 1 & 2 are different from MPH because of how you turn the yaw up, down, left, and right. In most FPSs, one analog/mouse/joy stick/etc. is supposed to turn all too all angles, while the other buttons/keys/etc. are supposed to move your position forward, left, right, and backward. MP 1 & 2 use a different style than that-- One analog stick moves the angle left and right, and when you push it up or down, your position goes forward or backward.
I tried to explain it the best I could, so hopefully, you all understand. But on some things, I may be wrong. This is from what I've heard.
My favorite FPS is Half-Life 2 for PC. The mod for it called Garry's Mod is very fun. All you do is mess around with the physics and with endless possibilities for making contraptions.