I'm honestly amazed that the whole 'what is the definition of gamer' debate keeps coming up. It's one of those charged words that means something different for everyone, and as such is essentially worthless without that context of personal meaning.
My own definition of 'gamer' is a hobbyist who has a definite interest in video and/or computer games. As these free-to-play games become ubiquitous around the internet, playing them doesn't create any particular distinction. I recall a while back here on the Escapist - I believe it was an actual article, but I don't remember exactly - a comparison was made to movie buffs. Someone who sees a movie every so often isn't given any particular distinction, because virtually everyone watches movies - it's only when they treat their movie watching as a hobby, showing interest and making a significant investment in watching movies, that you'd call them a 'movie buff'. The same thing applies to games - someone who plays free online games at work just to kill time, or picks up a new game every few months, isn't a gamer by my definition, because they're treating it like just another way to kill time. It's at the hobbyist level, when that significant investment of both time and interest are made, that the distinction has any meaning.
So no, as 'a person who plays games' is no longer out of the ordinary, it is no longer enough to warrant the distinction of 'gamer', just as 'a person who watches movies' doesn't have a special name.