I live in Ohio and my father needs to go to a nursing home.

Huh? My mother, on Medicaid, had money set aside for her "funeral." She had an arrangement with a funeral home.  She did not have a "burial contract". She wasn't buried -- she was cremated. Her funds covered her memorial service.

Maybe they just mean that in addition to making the funeral home the beneficiary you also need some written agreement that they will use the money for your mother's needs. Being beneficiary does not mean they can use the money on remodeling the viewing room! 

I think you need some clarification from the person at Medicaid. Maybe even a sample form you can show the funeral director.