Only true scotsmen can be true gamers!
You know you have a problem in your argument, if you need the qualifier 'true' without ever really defining it. In this case it means: you are an old-school gamer (as I am by the way) and you have seen yourself as a gamer for years and now your hobby has extended to other groups, but you don't feel familiar. So you say: these may be gamers, but I am a true gamer! In this regard your point 'true gamers play every system' is quite telling: you name gaming consoles, but no flash games, no mobile games, no text-based console games, no MUDs. So, are you yourself according to your own definition not a true gamer? Or does a 'true gamer' only need to play on 'true gaming systems'?
There are better qualifiers than 'true'. Console gamer. PC gamer. Mobile gamer. MOBA-player. Shooter player. And so on. Why not stick to these, as these don't try to classify and give a value judgement (as 'true' has a positive connotation) at the same time.
I am myself an old-school gamer. For a long time I never touched a console. I played on DOS: Monkey Island, Indiana Jones, Doom, Duke Nukem 3D, Dynablaster (european name of Bomberman for PC), Micro Machines, XCOM:UFO, Civilization, Colonization, Return to Zork, MUDs, Pinball Dreams/Fantasy, Starcraft, Dune, Baldur's Gate, ... I started to grow away from the gaming in the 2000s, but felt unsettled. Then Nintendo roped me into console gaming with the Wii. So my experience is a complete different one from yours. If I talk about old-school it is something completely different than for you.
As an example: you say a true gamer doesn't care about frame rates. As old-school DOS player I know that many games back then hade a simple code to show the frames. And I saw frames in the hundreds. Not always. But it happened. The discussion over 30fps vs. 60fps feels off for me, as I saw much higher frame rates. That is obviously because the PC platform had wildly differing specs and moved back then pretty fast. Even if you had the game two or three years later your PC usually could show the game with excessive high frame rates. Or the game was designed for VGA or SVGA and you had a Voodoo card or something.
My point is, that gaming is pretty diverse. And a mobile gamer or a player of flash games is as much a 'true' gamer as a console or PC gamer. And in the future the Stadia players will be as well. As long as you play games and have fun with it, you are a 'true' gamer. If you want to group gamers into different buckets, you need more specific definitions.
