My understanding and guess

May not be accurate. Just my reading of the fine lines.

When one rents a vehicle, the renter's own insurance policy covers liability. When the renter got into an accident and was at fault, the renter's insurance would cover the other vehicle's damages.

When one rents a vehicle and purchases the insurance from the rental company, that insurance covers damages to the rental vehicle as secondary insurance. When the renter got into an accident and was at fault, this insurance would cover damages to the rental vehicle.

In this case, it sounds like you were at fault in the accident and your own auto insurrance needs to pay the other party's damages.