i believe the center of mass is inside the sun if you look at each planet-sun pair individually. IOW, 2 body to n-body kind of messes this definition up.
also, by inside the sun do you mean inside the corona?
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.
the event horizon (schwarzschild radius) of the sun is 3km.
i don't think planethood should be defined based on whether the barycenter of the system is within the orbited object, but I do think that's a pretty good definition for determining whether two objects are a planet moon pair or dual planets.
but in case you con't want to follow, here's a summary:
dual planet system (eg pluto charon): barycenter lies outside of either object
planet with moon (eg earth and moon): barycenter lies within the radius of the planet
planet: "(a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit[1]." [2]
as for why planethood shouldn't be determined by barycenter location alone:
"If the definition of a double- or binary-star system is used as a comparison, and it depended only on the location of the barycenter, then any revolving body with a barycenter beneath a star's surface would be a planet, and any body with a barycenter lying outside the surface of the star would be another star. In the Solar System, all of the major planets would be planets under this definition except one. The Sun–Jupiter barycenter is the only center of mass that lies outside the surface of the Sun. Therefore, since Jupiter is not a star, the difficulty faced by astronomers to derive a reality-based definition of double planet begins to become clear." [3]
[1]: meaning it has become gravitationally dominant, and there are no other bodies of comparable size other than its own satellites or those otherwise under its gravitational influence.
also, by inside the sun do you mean inside the corona?