Playing "thirsty baby" is when a man instinctively tries to suck on the boobies of the woman he's with. This can happen anywhere at any time, asleep or awake, and the man may or may not know the woman. The relevant part is the random, instinctive act of boobie sucking.
You're in bed with your woman and she's asleep. You roll over and start sucking on her boobies. It doesn't matter if she's awake or even enjoying it. That would be a text book Thirsty Baby.