Everything orbits a barycentre. The position of that barycentre within or without the primary is not really relevant within several orders of approximation—and when it does become relevant, then that system becomes interesting in itself. Chances are the planet in question is so massive relative to its primary that we need to ask a more fundamental question: are we even looking at a star-planet relationship, or is it closer to a binary star system? Then we can investigate how that system formed and developed, and generalise and improve our model of stellar/planetary system development.
Yes — their point is that as-written, it orbits a barycenter external to the star: hence doesn’t orbit the star, but they’re in a mutual orbit.
Thus the suggestion for a new dwarf category — precisely because we do label times the barycenter is outside the star differently, as you yourself note.
You were so busy trying to “correct” me, you failed to address the topic.
There are plenty of binary star systems where this orders of magnitude standard is not met .
There is no academic literature or accepted consensus barycenter has to be orders of magnitude away - the accepted consensus is that a body orbiting another means internal barycenter, if they have external barycenter then they orbit each other period.
The fact that Charon and Pluto do not have internal barycenter is why Pluto-Charon (also Eris-Dysonmia) are considered dwarf planet systems ; and that was used partly in support (while not used in definition) of arguing for dwarf status of Pluto .