How can the definition of a planetary satellite be used to question Jupiter’s position as a planet?

Question

With all the outcry about Pluto no longer being a planet I thought of this odd point I have seen no one else bring up yet. Wondering if anyone else at all has picked up on it. Had Pluto been defined as a planet then its companion Cheron would also have been classified as a planet? How could/would this double planet definition possibly cause an issue with Jupiter being a planet?
(Ok I probably gave to many clues in the additional details)

Answer 1

The center of mass of the Sun-Jupiter system is outside the surface of the Sun, meaning Jupiter could be considered a binary companion with the Sun, rather than a planetary satellite.

I found that surprising.

Sun's mass: approximately 2 x 10^30 kg
Jupiter's mass: approximately 1.9 x 10^27 kg

distance between center of Sun/Jupiter: 775x10^6 km

Center of mass: 736,000 km
Radius of Sun: 700,000 km

Answer 2

I think that 'double planet' or, in the case of Pluto 'double dwarf planet' is still up for debate.

One simple way of defining a planet with moons over a double- or multiple- planet system is to take any planet-moon system with the centre of mass INSIDE the planet to be a planet and outside to be a double planet.
Not sure how this would affect Jupiter as, although its moons are fairly large, they're dwarfed by the planet itself.