Yes, but those are typically for games like MMOs where the servers are responsible for a large number of players. Those are never presented to the user as "your

Yes, but those are typically for games like MMOs where the servers are responsible for a large number of players. Those are never presented to the user as "your server," they're "the game's servers." Contrast with Discord where there's something of an illusion of ownership; people will refer to them as "my Discord server." EDIT: Incidentally, if you're actually running a private server for an MMO, I would indeed call that "your server."