When you mention Marriott will only give rewards if booked through their website, I'm assuming you mean no points when booking through other travel vendors or agents as opposed to just calling the hotel or marriott reservations line. I've gotten points by booking directly at the hotel or through the reservations line also, in addition to their booking via their website. Don't want to speak for you but I believe you meant to say when booking with Marriott directly. I've also been able to add my MR number at check-in if someone else made the reservation for me.

I actually think points reimbursement is more related to paying for a certain rate, like rack, corporate, government/military, etc. Most programs terms of conditions specify they will not award points for discounted rates, of which priceline sells.