As an Air Force Reservist (yes, I left the Army for the Air Reserve because they have a/c and cable), direct patient contact personnel are usually referred to by their job titles or associated nicknames (i.e. I'm referred to as a BeE because my job title is Bioenvironmental Engineering), LPNs and RNs (enlisted and officer) tend to be referred to as nurses, flight medics (I'm not sure what the official term is) tend to be medics - these are the NREMT certified basics that are the baseline of military medicine in all branches from what I understand. A friend of mine has a son that just got through corpsman training and he's NREMT-B certified now. Anywho, all that being put aside, the common terms I've heard in my limited exposure to the Air Force for personnel are the official "Airman" term, medic or flight medic in reference to medical personnel by non-medical personnel, and "trooper" as an all encompassing term for AF personnel in lieu of airman. It's not always easy to identify who is patient care personnel by uniform insignia alone. The AF uses a job field identification type badge above the name tag on the left breast, but the snake and staff only broadly identifies the personnel as working in the medical field in some shape form or fashion. EFMB and CMB badge wearers are not necessarily Army medics. Personnel wearing medical unit patches also run the range of job fields. I recall the Navy having similar insignia for veterinary services personnel as well as corpsmen (though it's been a long time since I've worked with any of them).
I guess what I'm saying is the best way to figure out who's a doc, nurse, or medic as far as the military and military type services are is to observe them carrying out patient care or getting a good look at their CV. It's just like the civilian side - I've seen a lot of NREMT certified EMT-Paramedics that I wouldn't trust to feed my fish - titles and nicknames only give you part of the story. Working a few hundred hours with them will tell you who's a real medic.