what is the official term for "baby catcher"?

MidWife

Parturition quarterback?

At our hospital its basically the nursery nurse that transitions the baby from birth to going out to mom....then we have post partum nurses that are called mother/baby nurses. Although often if the nursery nurse is busy, the L&D nurse becomes the baby catcher. The one grabbing the baby from the doctor.

Midwife, Doctor, Obstetrician, --the one who delivers the baby.