That's right and that is making a game medium for gambling because of the bets that you can made for each game that you will play. And in combination of both you are going to enjoy a game and with the chance of getting profit if you are will going to be that lucky.
I don't really agree with this as when you say game, this is prettyuch a general term. There are a lot of games where you can earn that is not considered gambling. For example, players under contracts get paid. They play a game with monetary benefits. Gambling is a kind of game that uses chance and luck to pretty much determine the result most of the time. So with what you said, it's the other way around. Gambling is a medium while game is the general classification.