Was the dealership correct that my daughters could drive their loaner if just my name is on the loaner paperwork-??

I have both my daughters on my car insurance. I have a joint car lease with one of my daughters -Lisa. A free loaner was promised with our lease if our car goes into the Service Shop or the Body Shop.. Unfortunately we had to go the Body shop. The dealership uses a rental agreement as a document to record the free loaner. I am a senior with osteoarthritis and my daughter has to often drive instead of me on my bad days. The dealership was told that. The back of the rental agreement states that only the customer or spouse, or the people named on the front of the agreement can drive the loaner or we breach the contract. There was room on the agreement for additional drivers to be added. The dealership put only my name as customer and said my daughter's name was not necessary. We went back and forth on this- until it got ugly – they said we were stressing them out. They finally added my daughters to the agreement. Through this whole ordeal they kept saying my name is only necessary on the agreement and both my daughters can drive the loaner. We live in Michigan. Were they telling us the truth??