For the first few levels of Gamer, the game system matters quite a bit. Be it so you can collect 'em, min/max them, abuse them, or complain and contrast them. These levels tend to be an adrenaline filled ride, and quite a rush.
After Gamer level four, you start to get access to the skills which suggest the rules themselves aren't as important as you thought. And maybe you start to doctor up your own set of house rules errata, or start to blend aspects from various systems you like, or just start writing up your own.
Around Gamer level seven, the social and creative aspects of gaming can come into sharper focus. This also tends be around the time of the realization that the raw supplies for gaming aren't just coming from RPG and office supply companies... but rather from life itself. Creative inspiration can suddenly be found almost anywhere, not just from books, movies, and songs, but every cultural medium... every thought or emotion.
By level eleven (or sooner, from certain types of cross-class synergy) you tend to have open access to the skills that let you liberally apply your gaming experience to manipulate many of the rules found in life itself.
And since I'm here, I'd like to give a big shout out to those who gamers who breeched the teen levels. Your secrets remain safe with us.