embassy births are US territories.
just as are all us territories... islands, governorships, military bases, embassy offices, clinics, treatment centers etc.
if your parents are staff at embassy or other us territory or jurisdiction and they are us citizens and YOU are born to them while they are NOT in the continental usa, but are on a base, embassy territory or property... YOU ARE AN ELLIGIBLE citizen.
Diplomatic staff in the U.S. live "on the economy" and their children are born in U.S. hospitals, not in an embassy compound.