Payton pounces into the bed for morning snuggles too. But he just jumps up on the bed and then presses himself against my legs... he's very unobnoxious about it. No asking for belly rubs or wookie noises or anything of that sort.
Auggie doesn't snuggle in the morning (or much at all really)... he just doesn't move at all until he knows I'm actually getting up. I can slap the snooze alarm two, three, four, five times... until I'm for real getting up, he doesn't move. He just lays with his head on the pillow next to mine. I don't know how he knows the difference between "getting up to hit snooze" and "actually getting up" - he just does. Payton does too, he won't get back out of the bed until I'm getting up for real, but I think he takes his cues from Auggie. Once Auggie gets up, he knows I'm actually getting up and jumps out of the bed.