Nope, 'Gamer' pretty much covers it.
The problem is when you DO need to sub classify, then it gets into a massive grey area about differentiating hardcore, mobile, web, mmo, e-sports, casual gaming... etc. etc. For some reason some people get real defensive about their particular brand of gaming (*Cough* PC Master Race *Cough).
It's weird we feel the need to classify it at all, I don't consider myself a 'Film-er' or a 'TV-er' yet I do these things also.