That's a rather unsatisfactory way to define moon-hood, because it doesn't depend on any essential features of the two bodies. Replace Earth with something of the same mass but more dense, and the Moon gets promoted to planet, even though its size, shape, and orbit don't change? Yuck!
I believe the center of mass for the earth moon system is something like 3000 km from the center of the earth (in a moonward direction). Just for reference, the radius of the earth is 6300 km.
The density of the earth is ~5.52g/cc, and halving the radius of the earth will increase the density by 8 which gives a density of 44.16 g/cc or 44,160 kg/m^3, which is about twice the density of osmium and a third as dense as the center of the sun.
In other words, the scenario you imagine is not practically possible in a system where one body is much larger than the other (if they're both about the same size, then it a dual planet system).
I don't think this destroys your argument, but the barycenter of the Earth-Moon system is located, according to wikipedia [1], 4670 km from the center of the Earth.
Interestingly, both the Pluto-Charon and the Sun-Jupiter systems have barycenters above the surface of the primary body.
pluto-charon (imo) is a dual dwarf-planet system not planet and moon.
Sun Jupiter is my argument for why a planet shouldn't be defined only by barycenter location.
updated density: 13.55g/cc.
this is about the current density of the core of the earth now. It's about halfway between rhodium and mercury at STP, so not as horribly unlikely, but still pretty out there.
> Sun Jupiter is my argument for why a planet shouldn't be defined only by barycenter location.
But Jupiter doesn't sustain nuclear fusion so it can't be considered in the same category as the sun. If it was a red/brown dwarf, then we would be in a binary system and that'd be a different story entirely.
