But that doesn't address my main point, which is that there were cultural changes and evolutions in other places - in some, greater than the changes that took place between the early Eastern Roman Empire and medieval Eastern Roman Empire - yet we have no problem calling them by the same name. We examine a polity that continued to exist in an unbroken line for centuries and say, "Well, things changed over the years so we're going to refer to it by a term that was never used to describe it contemporaneously."
If a distinction needs to be made, it's between Ancient Rome and Medieval Rome; there never existed at any point in history a man called the "Byzantine Emperor".
The problem is that the pendulum is maybe goiing a bit far on other side - while 'Byzance' is not Gibbon's 'travesty', MUCH changed more than said maybe, and evolved, so it's not anymore the empire of old on a few levels.
Underplaying those changes.
Again, Romans maybe, yeah. But Latins, not. And the slow evolution, changes like heightened cesaropapism and all... well...