Mission range is the same as operation radius.
I don't think that HEMS helicopter have to search for anyone, than hover over them to lower a rescuer, and then pick him and other persons up. It's 60 miles, but at that 60 miles mark you have to do a job that will use up quite a lot of fuel. That range is just a half of the capability you have on the remaining fuel after taking what you need for the job. HEMS uses way less fuel doing the job on the spot so the range is bigger.
So if the mission range is set expecting the job to take 2 hours on the spot... the resulting range will be quite small.
I would guess it is always a mix of both "operational thing" and "limitation of the aircraft configuration". The machine is suited for the job (equipment adds weight, so it takes fuel) and the foreseen operation (sometimes worst case scenario, sometimes a statistical average) will dictate how much of that fuel will be used for job. Whats left is for stating the range that job can be done at.