I don't think a "Gamer" relates to playing a certain game. There are just too many genres to limit it to certain titles.
That being said, if somebody hasn't played Zelda (As listed above), Elder Scrolls, Pokemon, Final Fantasy, or any of the big names, I would begin to question their choice in titles. Many people do prefer FPS like Halo, Call of Duty, and Battlefield. For strategy, I'd say League of Legends and DOTA are big names.
Like I said, a lot of it is personal preference. I'm not a fan of FPS games, and although I love a good Role-playing Game, I've never played WOW. I don't like paying by the month, so I don't really play MMOs or things that require a subscription.
I tried WOW a long time ago and did not like it when you first got on the game others would be waiting there to attack you, kill you and then take all you had. I have heard they changed that a lot but that showed me that the developers did not care at the beginning so it was not for me.