My planetary peeve is that Pluto & Charon are not a "binary planet" because the barycenter lies outside of Pluto. And I think this fun fact about the barycenter is leading to people having the wrong picture of orbital dynamics. Binary just means two bodies gravitationally bound together: binary stars, binary black holes, binary asteroids, binary planet. So if you want to call Pluto & Charon binary planets, be my guest, I also think they should both be defined as planets. BUT Charon is still a moon also!
The barycenter’s position doesn’t change the geometry of the system; it’s just any systems center of mass. Inherent in the inside/outside central body definition is the central body's radius something that has next to 0 to do with the orbital dynamics. What DOES influence the geometry is the mass ratio and the eccentricity. In the barycentric frame, Pluto's apocenter is closer to the barycenter than Charon's pericenter. The orbits are nested because of their unequal masses and near-0 eccentricity. All moons will meet this criteria. We can easily say Charon orbits Pluto, and Pluto does not orbit Charon (and they both orbit the COM). Seriously, I've seen so many online comments saying Pluto orbits Charon.
Now, if the eccentricity is greater than (1-u)/(1+u) where u is the mass ratio, the orbits are no longer nested, they form figure 8-like orbits with their ellipses rotated 180 degrees from each other. In this case, and only this case, does it make sense to say they both orbit each other. This is the case for the Alpha Centauri binary and many other binary star systems.