That’s exactly what it’s for, and the reason there’s a display name and a name is specifically for these privacy concerns and so people don’t have to show their name to others. However, I’m not reproducing the issue you mentioned. I just created a new account, and with a new subscription, the game name was actually set to the display name of the user, not their name. And joining someone else’s game, the display name is what gets displayed as the player name on hover.
You are not the first one to complain about this however, so I’ll continue investigating in case there’s something I’m missing.
Do note however that the default behavior was that you joined a game with your display name as it was set at the time you joined (or created) the game, so if you changed it after you joined a game, then it would have remained set to the previous display name you had (which would have defaulted to your name when you created your account). This is because you could be in multiple different games and you’d want a different display name per game (usually set to your character’s name).
This default behavior changed in the update of November 13th, where now, it will keep in sync with your display name until you manually customize it.
You also made me realize that having the display name defaulting to your name when you create an account was not a good idea, so I’m changing it to default to the username you chose. I’ll be pushing that update within the hour.