I think this just ends up being confusing. For games where player 2 hasn't joined yet, saying that you're in "round 1" might lead folks to think they can make moves (they can't). For games that ended early (e.g. player 1 wins the first two rounds in a three round game), seeing that it's round "2 of 3" leads to some confusion about whether or not the game is actually done.
If I do change this up, though, I might need to change it up for completed games: it's interesting to note when you beat the game early. Maybe a "John Doe won this X round game in only/exactly Y rounds." Something to think about.
I think this just ends up being confusing. For games where player 2 hasn't joined yet, saying that you're in "round 1" might lead folks to think they can make moves (they can't). For games that ended early (e.g. player 1 wins the first two rounds in a three round game), seeing that it's round "2 of 3" leads to some confusion about whether or not the game is actually done.
If I do change this up, though, I might need to change it up for completed games: it's interesting to note when you beat the game early. Maybe a "John Doe won this X round game in only/exactly Y rounds." Something to think about.