I'd not worry about some fixed universal true meaning of "game" in Plato's World of Ideas.
Being of "language game" (LW) and "World Game" (RBF) heritage, I'll obviously come with my own spin, but so does everyone "mean" (i.e. "spin") in some way.
Are maths "really" but language games? But what is "language" here? "Forms of life" said LW. So it just keeps on going. One spin after a next.
Follow what beaten tracks you will, or go off beat. Many games to play that way too. Be unusual. It's OK.
From this Reply-Tree at math-teach