My BSH Daisy does this as well. Of an evening she's started jumping on to my side of the bed and won't let me in until she's had her goodnight belly rubs. I've had to start going to bed earlier to make up the time!
My sister's cat that's become the family's cat does the exact same thing. He'll do the side flop, and if he isn't getting adequate attention he'll begin to wriggle really adorably.
I think that is ragdoll thing. My ragdoll always begs for belly rubs when I leave the room. But does not sit on a lap. She likes being around people but not on people :D
I'm not sure of her breed. My sister found her on the side of the road when she was only a few weeks old.
I always guessed she was a snowshoe, which is a Siamese mix breed. Her pattern is near identical to what comes up googling "snowshoe cat". But I can't be certain.
Had a gray tabby named Dax that would fall asleep to belly rubs. We picked him outside a convenience store after he was abandoned there, we did whole found posters and everything but they kept getting torn down so after 30 days he was ours. At first he would claw and bite at any petting but we were able to get him to stop by pulling your hand away, as safely as possible, and giving a stern 'no' followed by ignoring him for a few minutes. He picked up that if he wants pets he can't claw and bite anymore.
My (neutered) boy, Silky, absolutely adored tummy rubs. Not just chest, but actual lower abdomen. He'd purr his head off and streeetch (and drool, after FORL destroyed most of his teeth). As far as I know he was just a regular moggie - he certainly didn't come with papers from the shelter!