Since I do a lot of mileage annually, and since I am a paramedic, I've been the first person to come across a number of wrecks and render aid over the years. I always identify myself with "Hi, I'm Jeff, and I'm a paramedic . . ."
I've never volunteered my full name or department affiliation, and won't give it unless specifically asked by a law enforcement officer, which has never happened. Of course, I'm fairly well known in the local fire/rescue community as an instructor, lecturer and attorney. But my usual approach is to just give a brief verbal patient report to the first arriving medical responders, then quietly back get back into my car and leave.
Jeff