Nurse Dragon
A Nurse Dragon is any dragon, male or female, who is part of the Dragon Corps medical services and thoroughly trained in the medical and healing arts. Nurse dragons handle all medically related duties within a weyr, and oversee not only battlefield medicine, but also the hatching of eggs, the care of younglings, the maintaining of the weyr's rookeries (where eggs go to be kept safe until they hatch), as well as overseeing the general health of all the dragons within the weyr. While generally small in total number, they play a very vital and important roll in daily weyr life, both in the weyr itself, and out on missions.
Battlefield medics, who are a subset of the nurse dragons, are dragons trained specifically in battlefield medicine, and who travel with fighting units into the battlefield, but act merely as support for the fighting units, but are never a direct part of them. As such, no unit has a single nurse dragon, or battlefield medic, permanently assigned to their ranks. The medics are assigned on a first come, first serve basis depending on the needs of the mission. If one is determined to not be needed for a given mission, then the dragons of the deployed unit will see to their own medical care, as they too have healing abilities the same as the nurse dragons.
They're just not as skillful or well trained as the nurse dragons. The nurse dragons also work very closely with the weyr's techops (think dragon computer nerds) who oversee the more technical sides of dragon existence, including the millions, billions, or even trillions of nanites that exist in a dragon's body, the total number of which depend on the age, and size of the dragon in question.