The way I have always understood it is the ambulance crew is in charge of patient care and the fd has extrication. If fire gets on scene first then they would provide ems untill the medics arrived to take over pt care.
The biggest thing that matters is that both fire and ems work together for the pt. Ems needs to be able to gain access quickly, assess and stabilize the pt, then monitor them through the extrication process. Fire needs to be able to extricate the pt as quickly as possible using the ems crews assessment of injuries and what damage the vehicle has to do the job quickly.
Both crews need to be able to work together while trying to not hinder the others work. At times extrication may need to stop to let ems stabilize the pt and at times ems may need to step back to let fire take care of extrication.
As long as both fire and ems can work together to take care of the pt and get them out of the vehicle and headed to the hospital asap, then everything should work out fine.
At our FD it is pretty simple, if we have to go into a vehicle for patient care the FD person with most medical training goes in. Where I am at EMS has zero extracation training and we do our level best to keep them out of a car. If there are any questions EMS and FF in car communicate and it seems to work pretty well. That way those doing the cutting are confident that person inside the car knows what they are doing and they arent going to place themselves in the way.
My department is pretty good with patient care.
Technically, according to SoP's, I am unsure; I am inclined to say that it is the Medic/EMT who has the most experience or knowledge OR the senior EMS officer OR the Fire Chief/Capt./Lt. thereafter.
My department works pretty well in unison, especially when it comes to EMS. I've never seen anyone throw a powertrip hissy fit or anything.
Here we have the same issues of who is in charge. I have always treated it like this. We use the rescue zones per 1670 and if you don't have you gear on then you don't come in my(extrication officer) hot zone. Don't matter if you are a doctor or not. EMS is patient care and extrication is fire or rescue squads responsibility. If we can work together on prolonged extrications then I try to accommadate EMS but if I know I can do a rapid extrication then I bring them the patient when we free them from the vehicle.
The extrication should be going along with the patient care if they have the right gear and are allowed in my hot zone. If the patient care stops my extrication then that is not acceptable to the operation. If they are truely pinned in then there will not be much room for any good patient care.
Incident Command is in charge and has a lot of determination on how the incident is run due to the liability that the position holds.
I haven't had this issue but the senior EMT worries about patient care and the senior fire fighter worries how to get the patient out of that crunched car. In my 24 years I've never had that problem. Maybe I'm lucky or that it's just an understanding of senior personnel or officers of who is in charge of what.