I think "Steve's law" is an improvement, and I'm happy to learn the word "barycenter." However, I don't think the concept of binary planet (and above) is meaningful. The term "planet" should refer to the important bodies of a solar system with minor refinements.
Pluto's diameter is 2372km with the barycenter 960km above the surface. Charon's diameter is 1208km with a semi-major axis (which I take to mean closest orbit) of 17,536km. So the barycenter is 95% of Charon's orbital distance away from Charon. Furthermore, the barycenter is within a planetary diameter of Pluto, whereas it's around 14 Charon diameters away from Charon. This is s severely lopsided relationship. Calling the Pluto system a double planet and putting Charon in the same classification as Jupiter waters down the definition of planet to the point it is almost meaningless. Therefor, I suggest we refine Steve's definition to say that a planet must have it's barycenter within the planet body. Keeping it within a planetary diameter of the surface is also acceptable.
I would add a fourth rule: a planet cannot be tidally locked. Pluto and Charon are tidally locked, so out they go.
Finally, I would add a fifth rule: a planetary orbit must enter the heliosphere. Remember when Voyager passed though the heliopause and was said to have left the solar system. If it's outside the solar system, it's not a planet. It's something else. The recently-proposed planet 9 is so far out that it never enters the heliosphere. It's something else. Maybe an epi-planet.