When you train, do you have a higher chance to gain more of a stat the faster you press the button after your alert pops up? It might just be confirmation bias, but it seems that if I go and 'finish' my pet's training as soon as the alert tells me they're done, I'm more likely to get 2-3 increases on the stat I was training as opposed to when I come back like an hour later.

I think it's random and doesn't have anything to do with how fast you press the button. When auto training you don't press any buttons either and it still rewards 1, 2 or 3 of a stat per session.

Sweet, thanks for letting me know!