If we're talking the dictionary meaning (no connotation) then a gamer is someone who plays games (PC or console). Therefore CoD players are gamers.
If you're talking connotative, this is where things get sticky. Using driving as an example isn't fair because, to be honest, driving is a necessary skill (most of the time) unless you drive mostly for the pleasure of it. If you can afford to do so and you don't absolutely need it, it means you're doing so as a hobby. How far into the hobby you go will determine the extent of your commitment to it. Calling someone a racer for driving around town would be inaccurate. They need to compete with other drivers, legally or illegally if they're into that sort of thing.
Hardcore gamers believe they're more racers than simply drivers. In other words, they would scoff at someone claiming to be a racer for drag racing their mom's Uno from one traffic light to the next. The professionals would believe their track racing or modded vehicle (in gaming terms, a wider collection of games, plenty of practise and great knowledge on the subject) trumps the Uno driver's small traffic light to traffic light drag. Technically speaking, of course, someone taking part in a race is a racer, but there is a distinction between the two.
The problem - if we're arguing about word definitions here - is that we have limited words for describing different degrees of gamers. Casual gaming is supposed to mean you love mobile gaming (and usually includes online games, such as those on Facebook and games like Zuma Deluxe or Candy Crush among others), but what happens if, say, you played chess online or on your phone? That's incredibly competitive and it would clearly place you in a different category to the "casual" chess player. And by the way, chess is just as bad. Knowing the moves and playing on the odd occasion doesn't guarantee you will be seen as a "true" chess player no matter how much you argue that you are (and no matter how many times you point to the dictionary meaning). I had a friend who competed, won national titles and even though I knew how to play, he never would have considered me a "true" chess player.
This is why it gets sticky. All we really have is hardcore and casual gamers under the banner of "gaming" or "gamers". In other words, you have gamers but people like to believe they are either hardcore or casual. Hardcore and casual, though, are two ends of the same spectrum. In between there are quite a few people who have an interest in playing games, but not necessarily any interest in spending their time building gaming libraries or learning about more than a few games. They're the Uno drivers who bought the car from mom, slapped on a turbo and often challenge their friends for a few bucks. Sure, technically they're racing and I know a racetrack nearby which encourages street racers to take part in Street2Strip events and keep their speeding on the track so they don't kill anyone on the road. They're not professionals (hardcore) and they're not just driving for pleasure (casual) but I believe they would like to consider themselves racers (or at the very least, drag racers).
Should drag racers be considered proper racers if they take to the strip? Do the racers who have sunk hours into track time have a point when they say the Street2Strip guys are not "true" racers? Therefore, should gamers who only play Call of Duty be considered different from more "hardcore" gamers who have sunk more (overall) time into the hobby and expanded their libraries and knowledge in the process?