When you stop at a car crash alongside the road, to render assistance, you're the first responder.
A medical crew providing ACLS and other advanced life support enroute to an advanced care facility isn't a first responder. There are occasions when it's possible they might be, but rarely so.
Some years ago I waited at a rural airport for the first victim of a multi-casualty incident. Our crew had been picked up by the ambulance and taken to the clinic. They were on their way back to be loaded, and transported. The ambulance backed up to the aircraft, and I could see a lot of activity inside. The doors swung open, our head medic shouted for me to start tossing him trauma bags and gear. They'd used up what was on the ambulance, and they used up what we had. I was pulled into the ambulance to help. The ambulance ended up going back to the clinic. The patient couldn't be stabilized for transport. I believe that person perished.
The person was a first responder. The person stopped for a car wreck, and while attempting to help, was hit and run over by another car.
We were not first responders, despite being mired in the situation.
The nurses and medics on many of these flights hold more qualifications and more letters after their name than can scarcely fit on a business card. They're exceptionally qualified and certified.
Air ambulance may be called to the scene, in the case of rotor wing, and scene transports typically are more involved in stabilizing the patient than fixed wing transports, but in nearly all cases, the patient has already received initial first responder care which may range from chest compressions to pressure on a wound, to basic live support, even intermediate or higher care by law enforcement, fire, paramedics, EMT's, etc. For fixed wing transport, the patient may be moving between clinics or hospitals, or maybe delivered by ambulance directly from the scene; in either case, first responders have already done their part, and the fixed wing crew is transporting and providing enroute care, but not acting as a first responder.
Anyone can be a first responder, and given human nature, in emergencies, often anyone who can help, does. Frequently first responders have little or no training, but do whatever they can until services or personnel with more capability or qualification can be brought to the patient. Sometimes the first responder is a fire crew, sometimes a cop. I was involved in removing the roof of a vehicle and unable to cut a seat belt. A knife was handed me from behind; it came from a trucker, a first responder to the scene who was standing by to help any way he could. People are like that, and first responders come in all flavors from kids to qualified doctors and nurses. I've even been a first responder when riding as a passenger on a commercial flight, and several times I've been joined shortly after by a doctor or nurse...who was also a first responder. The Flight Attendants were first responders. Once that person was handed off to personnel at the gate, however, we were done, and those taking charge weren't first responders. Circumstances, sometimes qualifications, sometimes setting, dictates what constitutes a first responder. Seldom are medical flight crews first responders, though the nurses and medics who fulfill that critical role can be involved from the outset, on occasion. Typically the flight crew are those to whom the first responders hand off the patient, or whom receive the patient well after the first responders have done their part.
The flight nurses represent a very unique and distinct subset of the care chain for that patient. Once that flight takes off, the only resources and personnel for patient care are aboard. No one else can be pulled in, no more supplies or equipment obtained. During ground transport, services can meet the ambulance on the road, and in the hospital or clinic, often there are backups, spares, and extra personnel and supplies. In the air ambulance, it's all on those nurses and medics. In the case of each place I've worked, I'd trust my life to those people perhaps above anyone else, and I'd trust the life of my family, too. Without reservation, it takes a special kind of person to be a flight nurse or medic.