This rule is for when both bodies are the "same" type.
There are three types:
Black hole -> Sun -> planet/moon
Two suns are a binary sun if they orbit a COM outside both. Same for two planet/moons.
A sun around a black hole is not a binary sun, but we've never named such a thing (except maybe galaxy).
A small sun around a large one with the COM inside the larger one should have a special name, but we never gave that a name either since we've never seen one.
A planet around a black hole is a planet, since it's not the same type as the other one.
There are three types:
Black hole -> Sun -> planet/moon
Two suns are a binary sun if they orbit a COM outside both. Same for two planet/moons.
A sun around a black hole is not a binary sun, but we've never named such a thing (except maybe galaxy).
A small sun around a large one with the COM inside the larger one should have a special name, but we never gave that a name either since we've never seen one.
A planet around a black hole is a planet, since it's not the same type as the other one.