I'll admit, it can be easy to feel like those who've come to gaming recently get to have it easy after the rest of us have gone through the hard times. The folks whose first pen and paper RPG is the new Dungeons & Dragons never had parents and teachers tell them their soul was in danger. Lots of gamers today have never played a game on a computer that could only display 16 colors at a time or a console with pixels as big as your thumb. That doesn't mean they're not real gamers though.
As for the types of games people play, classic arcade games were some of the simplest games around, but I don't think anyone would argue that Pac-Man or Centipede isn't a 'real game'. Classic BBS door games like Legend of the Red Dragon and TradeWars 2002 were precursors to today's social games, complete with energy mechanics. You can't tell me those weren't real games.
Conversely, there are plenty of more complicated board games that don't get associated with the 'gamer' appellation. You don't see chess players calling themselves gamers. Monopoly, hated though it may be by many, is a game, but you don't hear about gamers playing it. Almost as if the fact that these games have been widely accepted for their entire history is a factor.
Myself, I think anyone who considers themselves a gamer is a gamer. Calling yourself a gamer implies that you consider the act of gaming a part of who you are. It's more than just a thing you do, it's an element of your personality. To me, a gamer is someone who believes that games matter. That's all it takes.