I've had a bill of sale/transfer of registration for every horse I've owned, and if a seller isn't willing to do that, then I'd walk away.
There's no GOOD reason someone selling a horse shouldn't give you a receipt/bill of sale/registration papers.
OP, I suggest you stand your ground on the issue of the DETAILED sales receipt....
I've owned three horses, and each sales receipt has stated their approx. Age, color, breed, height and markings as well as the sale price. All were signed by the owner.
What these owners are proposing is highly irregular.....
Do you realize that without a sales receipt they can come back on you later and say you STOLE the horse??? And how are you going to PROVE to law enforcement that you paid them? Because cash isn't traceable, as Speed said.