Why is Charon a moon of Pluto?

By defintion, any natural satellite or celestial body that orbits a planet or smaller body, which is called the primary, is a moon.

Simple physics suggest that because Charon is proportionally larger (when compared to Pluto), the orbits of both bodies are going to be "a bit eccentric" or as you've written above, barycentric... but Pluto is still the primary as it significantly out-masses Charon. Add Nix, Hydra, P4 and P5 (all of which are much smaller, but still have some gravitational effects) and the orbital foibles of this strange family will only grow that much more interesting.

I hope this helps answers your question.

One aspect of the IAU's thinking, when they were working out the definitions, is that the barycentre of the Jupiter-Sun system is outside the Sun. Yet Jupiter is regarded as a planet, not as a (brown dwarf) star, so for consistency Charon is regarded as a moon not a dwarf planet.

Because Charon does not have quite as much mass as Pluto does. Technically Pluto -Charon is a double dwarf planet and the four small moons are natural satellites.

http://www.trbimg.com/img-4ffdcc76/turbine/bal-wx-...

http://articles.baltimoresun.com/2012-07-11/news/b...