I think a true gamer is anyone who fairly regularly dedicates some of their time for gaming specifically to enjoy it. As opposed to someone who just kills time with it occasionally while waiting for something else to begin.
Basically. I would change "to enjoy" to "dedicatedly", as I know people doing speedruns and such play the same game so much they very likely don't enjoy it at all anymore unless their results are good.
A gamer (let's get rid of the term true) is someone that plays games regularly/dedicatedly because gaming is an important part of his life.
Be it a single genre, system or game, if one is willing to keep playing and is thinking about it constantly, that's about it.
I agree with the point of using gaming to just kill some time for something else, that wouldn't be a gamer, but someone that plays games sometimes.
If someone owns a dedicated gaming system and keeps playing it (and not just get rid of it or just dust it after two months) than it's likely a gamer, even if all he have is still a SNES or MD to play Mario or Sonic to this day, if he is still playing he is a gamer.