This is one of the distinctions between HEMS and Air Ambulance
Confused now. I was always under the impression that HEMS was the generic term for the Air Ambulance services (charities)
Certainly in this part of the world the same airframe (G-HEMS at the moment) does both roles i.e. attends emergencies and is used for hospital transfers!
Most, if not all Air Ambulance charities do both HEMS and AA work. The distinction is in the service provided and in the exemptions granted to a/c operating in these roles. HEMS is where the a/c is providing a medical service by air (paramedics and doctors to the scene and then transport out) whereas AA is just the transport part where the a/c becomes an ambulance with a rotor, frequently used for transfers and transport from remote areas where a land crew may not be able to extract the casualty in a timely fashion.
The difference is that if you are landing on a HEMS mission (life or limb saving) the you can land in a 2D site. If you are on AA then you can't. there are other differences as well but that is how I was taught to interpret the rules.
Air Ambulances doing transfers - does the charity get any money? I think the ambulance service does but not sure whether the charity does or not