"Core" gamer was a term Nintendo initially used (mainly) to reference the 'Hard-Core' gamers ...
The types of things a "Core" gamer does is play a lot of videogames, understand the technologies behind games, talk about games (online), care about the sales of games and game systems, and discuss whether graphics really matter ... It (in reality) has very little to do with genre.
People complaining about Nintendo not making enough games to satisfy their needs for a particular genre are the most annoying people around. They complain that "Halo 3, Mass Effect, Bio Shock, Killzone 2, and Unreal Tournament 3" are all being released this year and Nintendo needs to answer it; if you're complaining about this odds are that you already have the systems needed to play these games so play these games and then we'll see about whether Nintendo should make a game for you. No game company can make every game in their library every year; any attempt to create many of these games will result in lower quality games and lower sales.