I put video gamers in the same category as board gamers or card players and since I don't refer to world ranked chess players or professional poker players as athletes, you can see where my opinion is coming from.
I can understand why competitive gamers would like to call themselves athletes, but the only relevance here is that there are varying levels of skill that elevate all the way to a professional, ranked level of play. The skills sets don't require agility, balance, body movement or any type of physical exertion that defines sports games.
Having the level of dedication required to be at that level is both uncommon and respectable, but I would not place a world champion Street Fighter or Mortal Kombat player in the same category as a professional boxer or MMA fighter. I certainly wouldn't call them both athletes any more than I'd consider a world class EA Sports NBA 2014 player an athlete same as someone who actually played in the NBA.