…because it sure isn't 20 rounds daily. Since there's no alert I forget to play more than two or three times a day, but I really want to know what the real limit is.
It's 20 rounds a day. There's no time limit between games.
20 rounds daily per active pet, iirc. There's no cooldown between games and (again, iirc) you can switch between active pets to play more rounds.
Nope.

I'm not asking the cooldown between the x/20 rounds I'm talking about the number of 20 round games you can play daily and how often they can be played. As you can see I'm using the same active pet and included the time, both were today.
I'm fairly sure that's a bug. It's not supposed to reset during the day like that.
Are you switching between active pets and somehow confusing it into resetting, maybe?