Really, the only categorization issue that I'm adamant about is that Pluto-Charon is called a binary. The Pluto-Charon barycentre is not inside Pluto, therefore Charon is not rotating around Pluto, the two are corotating around a common point of space between them. That's a binary.
That definition works, until you realise that Jupiter's barycentre with the Sun is outside of the Sun. Would you consider Jupiter and the Sun a binary system?
To be honest, I don't, but I don't have any real reason _why_ I don't. We could make some arbitrary percentage to define something as a binary, but that's nowhere near as neat as the barycentre not being inside the main body.
Also, if you define binaries as the barycentre being outside of the main body, you're arbitrarily discriminating against small dense suns and planets compared to large diffuse suns and planets. For example, just about any planet orbiting a white dwarf will have a barycentre outside of the white dwarf, whereas just about any planet orbiting a red giant will have a barycentre inside the red giant.