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 .
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.
The topic is pointless hair-splitting and nostalgia. It's like saying we can't ever use F = ma to describe free-body diagrams; we need a full relativistic per-particle QFT simulation to accurately and precisely model every single interation. That is impractical, at least for now. Likewise, we come up with classifications that are useful for the present time. If 'dwarf planet' is a catch all for all the small roundish bodies in our planetary system, then so be it. I suspect the IAU will make things more fine grained in the relatively near future. We even classify the asteroids as more than just rocky lumps.
We can similarly say that “cold dwarf binary systems behave similarly to planetary systems, due to the proximity of the barycenter to the star”.
And that’s more informative than calling Jupiter a planet.