So which type is a neutron star? :)
In any case, we do say that a "sun around a black hole" is a "binary star system". See http://www.sciencedaily.com/releases/2011/03/110325082725.ht... and http://chandra.harvard.edu/photo/category/blackholes.html for two examples of many web pages which use that terminology. From a caption at the chandra.harvard.edu site:
"A binary star system consisting of a black hole and a normal star, located about 11,000 light years from Earth."
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.