Let's just acknowledge that anything big enough to be round is a planet. That's the bare minimum criteria.
Orbit shapes and clear paths don't matter, the Solar system isn't a typical stellar system, many aren't so stable and ordered, especially in binary and triplet star systems. So the pedantry around the shapes of the orbits of the outer kuiper planets is a very silly thing to argue about. After all most orbits in binary and triplet systems aren't even predictable long term, let alone not circular.
I believe the rule of thumb is binary planets' barycentre is external to either body. This is the case with Pluto/Charon, I think it's also the case with Earth/Moon.
Other way around, the further apart the objects are the less likely the barycentre is to be inside one of them, you can picture it as a rubber band with a dot drawn on it, the more you stretch it the further the dot gets from both ends even if it gets further from one end faster.
That's a good rule of thumb... but it's probably not enough; no reasonable definition would call Jupiter a star, or even a brown dwarf, or the Solar System a binary system, yet the Sol - Jupiter barycentre is outside the sun... (the whole system's barycentre is sometimes inside the sun, but that's due to Saturn's, Uranus', and Neptune's pulls cancelling Jupiter's).
I'd call the barycentre thing a necessary but not sufficient requirement; a proper definition of double planet should probably also take into account other factors like the relative mass and density of the bodies, and their minimum and maximum distance.