Any flight (including transfers, organ movement and carriage of medical staff) can be done as HEMS, where immediate and rapid transportation is essential. Anything else is "air ambulance" and is a normal public transport task.
Our local HEMS charity does not charge for any HEMS transfers (or 'air ambulance' transfers, if the casualty is resident in the funding area of the charity). 'Air ambulance' transfers of non residents to other parts of the country are charged by the charity.