What's the correct way of saying an individual game (one game = pay one time)?
If you look on the Wikipedia article for slot machines, it seems that the term round is used in the context that you are asking about, whereas games appears to mean the types of game that various machines offer. Also, the Wikipedia article and other sites always talk about bonus rounds, but never bonus games.
You could call the individual attempt on a slot machine a pull.
You've earned five free pulls.