I've always assumed the term meant, "Someone that played games", without assuming that it specified what sort or was exclusive to a subdomain. If I was asked to be more specific, I'd say it means, "Some that plays games .... a lot."
Certainly, when I hear the word "gamer", my bias is that is a person who shares my values with respect to games and plays video games, RPGs, CCG's, board games, and well pretty much anything with 'game' in it with almost equal relish. The more broadly you play games, enjoy games, and explore the world of games, the more title I assume you have to the word and the more likely I assume you adopt the word as not more a description but an identity.
As for Umbran's etymology, the 17th century version of the word meant 'athlete' (one who plays what we'd now call sports) and while the modern word 'gamer' is the same word, my assumption would be that it has an independent derivation. It's a natural word to create to mean "someone who plays games". You wouldn't need to know it from prior usage to adopt it casually.
That being said, gamer to mean specifically, "one who plays video games" isn't attested until 1993 which to me seems startlingly late. I could have sworn it existed in my vocabulary to describe RPG players before that, though I could well be misremembering, so it must be the press took a long while to catch up. I do recall prior to 1990 more hearing the term "dungeonhead", formed as a cognate of "metalhead", to describe the particular high school click of outsiders defined by their gaming, and sometimes self-adopted the term. But I'm pretty sure that "gamer" was on the tongue, and that for example, "wargamer" or "power gamer" was around and so "gamer" without the sub-classification was as well. However, if I was using "gamer" before 1990, I wouldn't have assumed that it specifically meant "one that plays RPGs". I wouldn't have thought to differentiate.
