Yeah, and here's what your wiki link states. Sport is generally recognised as activities based in physical athleticism or physical dexterity.
Sure, it is "generally recognised", i.e. by the masses. Read down to the definitions and make up your mind then
Wiki isn't the best place to find information. Lets try the Oxford dictionary..
British Version
http://oxforddictionaries.com/definition/english/sport
American
http://oxforddictionaries.com/definition/american_english/sport
I think I want to weigh in something else here that hasn't been brought up yet...
While I would agree with the idea that chess is not to be classified as a physically exerting competition (and thus, why many won't call it a sport), whether something is or isn't classified as a sport also involves what mindset the competitor is in.
Shift gears for a bit and consider skateboarding. It is definitely recognized as a competitive sport, and is definitely physical. Yet, if a teenager is skating down the street, or even trying the occassional kickflip off a curb, would we really say that they are training for a sport? It depends upon how serious they are taking it. There is an ill defined line as to when a skater is just playing around with their board in their driveway, and when they are training and trying to refine their skills for competition.
Back to chess, there are definitely people out there who only consider it a game because that's the only way that they have played it. They are not considering that there is a distinction between those who play it like a game, and then those who are training and trying to refine skills for competition. Competitive players definitely have a sportsman like attitude towards their matches and competitions. In that respect, yes, chess is a sport.
As said in my previous post, I think that this debate really just reveals how limiting and ill constructed the definition is of the word "sport". It is not all inclusive of what it means to be involved in serious and demanding competitions, physically exerting or otherwise. If you encounter someone who refuses to recognize chess as a sport, you might get them to concede something more important -- to achieve at chess, one must have a sportsman like attitude and devotion to the game.
a kid skating down the road or perhaps someone riding their bike to work/school is not a sport. It IS a physical activity which allows it to be turned into a sport when you add competition.
I can train all I want at something competitive and that doesn't make it a sport. If you think chess is a sport, then you would have to classify other games as sports... Poker, Starcraft, CoD: Black Ops, Farmville, World of warcraft etc... all competitive GAMES but nobody is going to call them sports.