The Definition of a Gamer
A gamer is a person who plays games. Period.

Now someone really deep in the hobby will try to play as many great games as they can, regardless of image. People deride Bejeweled and Kingdom Hearts, but I like both of them, as any gamer who are ok with Puzzle Games and Action RPGs would. I myself am not good at Stealth or Music Games, but I recognize that Metal Gear Solid and Rock Band are pinnacles of their genre and deserves their fanbase. Not every gamer has to play every genre, but they should respect people who do play genres that they don't like.