What defined it? People. Primarily with their reaction to them. Otherwise, it is just a disc.
Oh you want a specified game?
NO! you get 3.
Birthed it? Doom. Enough said.
Defined it? Halo (specifically 2 when console multiplayer became widespread and increased the popularity of shooters across the board)
Redefined it? Bioshock. Proper shooter with hybrid elements of RPGs with in depth and cohesive story and char building it has advanced the genera as a whole, because some people who wont play a shooter, will play bioshock willingly.
EDIT: you could argue that Kill.Switch redefined it, with the advent of the cover system, or gears of war with perhaps the fluidity and need to utilize the cover system. I personally wouldnt, but it is a valid argument.