Heer Sanitäter - Hilfskrankenträger (medics)

Can someone clarify a few things for me, I understand that the squad "medics" were normal members of a company with a little bit of advanced first aid training and when a combat situation made it necessary they "dropped" their weapon, got a red cross armband and a basic first aid pouch and provided basic first aid (bandages etc...) and these were known as Hilfskrankenträger (auxiliary stretcher bearers). A Sanitäter (sani) would be a member of the army medical corp and not an active combatant. Is this correct ?

I presume both a Hilfskrankenträger and Sanitäter would carry a medical pouch (instead of an ammo pouch) but would both wear the medical personnel (snake) patch ?