Difference of opinions here I guess but the definition I offered clarifies that the object to be considered a planet in any sort of clustered central orbiting system, has to be the most prominent body in that orbit. That would make Pluto the planet, and Charon the moon. Charon is about half the size of Pluto, so there is a significant difference on the gravitational influence of each celestial body as well.
The Earths moon is roughly a quarter size of our planet and in many respects, we share similarities to the Pluto-Charon system. I would argue that the Earth-Moon system is also a binary system too. However, the most prominent body of influence should retain the most prominent hierarchical name. Hence, the Earth is a planet, Luna is a moon, Pluto is a planet, and Charon is a moon.
Yeah I understand what you mean. And I understand that Charon kinda feel like a moon. But what if both (Charon and Pluto) were about the same size? You'll say that the one slightly bigger is the planet and the one slightly smaller the moon?
In my mind, for a Moon to be considered a "Moon", it have to be greatly smaller than it's planet. That's why I love to use the centre of orbit (or Barycenter) as a reference. If a "planet" is massive enough so it's clearly the "Master" of it's own system, then the barycenter will be inside itself.