Been doing some more research on this, and it appears that in English the word 'play' and the word 'game' are in final stages of almost completely reversing meanings.
Game comes from old English gamen meaning 'to be merry, to seek amusement, to have fun'. The word suggests unstructured leisure.
Play on the other hand comes from old English plegian meaning 'to act a role, to perform, to exercise'. The word suggests structured non-work activities - for example playing a piano, acting in a play (the noun form retains its original sense).
But now if we were to say that a child plays, we would mean that they were generally making merry and having fun in an unstructured way.
And if we were to say that a child games, we would mean that they were acting a role, engaged in a contest, or exercising in a structured fashion.
The two appear to have been sliding past each other since about 1300 when the stickiness of the word 'play' to the idea of 'theatrical performance' forced the word 'game' into the sense of 'a contest with rules', but I think the modern usage of 'gamer' just takes it one step further. Prior to 1300 or so though, if you children were running around outside screaming and generally behaving chaotically, you would have said, "The children game." Whereas, if the children were playing tag, or marbles, or pretending to be knights, you'd say, "The children play" or perhaps, "The children have a play of tag." or "The children have a play of knights." Additionally, 'player' seems to have more carried the sense 'gamer' has now. You can still see this in relationship to athletic games (now more often called 'sports') were we call the participants 'players' as opposed to gamers, even though they play a 'game'. 'Sport' comes from the old French where it basically meant the same as game did prior to the 1300 and appears to have replaced it as a verb for while, hanging on to the sense of general amusement seeking and making merry (and engaging in sexual activity because well, French) until the 16th century, when it started to have its current meaning as a noun. Sometime before that, the fact that hunting was the (second?) most popular sort of sport (in the sense of amusement or diversion) among certain classes caused animals killed in sport to become "game animals" or simply just "game". If you could hunt them legally, then they were "fair game".
"Gamer" was/is also an adjective - "more eager, bolder, and willing" That sense meaning courageous, however came to English out of "gaming", specifically betting on the fights between game cocks. That might explain why the usage is increasingly obsolete except in phrases like: "Are you up for it?", "I'm game." But in the early 18th century, if you said someone was into gaming, or a gamer, you probably almost certainly meant he bet on some violent contest between animals bred for that purpose. If the word had kept this sense, it would have almost certainly made the phrase, "Game Boy" ill-advised marketing.
I always feel a bit sorry for anyone that has to learn English, which is pretty much everyone actually. We are never satisfied to have just one word for something and keep it that way.
