We were told to call hospice first. They sent someone to examine Mom and pronounce her dead, and then they called the funeral home we had selected. From what I read above, that seems to be the regular practice. If not in hospice, then I would call 911.
Is the patient on hospice care? If so, I would ask them about it. My brother, sister, and I were there when our mother died. Hospice had told us to call them when that happened. We did, and they came and took care of a lot of things. They prepared a death certificate, asked us what funeral home we planned to use, then called the funeral home to come get my mother's body. Hospice was tremendously helpful.