I do not think that folks are disputing the mislabeling. Phase inversion is really only applicable to "inversion of a symmetric periodic" waveform (as stated above), with the key terms being symmetric and periodic. A real-world application is in AC currents where these can lead to catastrophic damage to breakers by connecting AC buses out of phase (which goes way beyond simple inversion). In an audio realm, "symmetric and periodic" is pretty much N/A, but can be "simulated" via polarity inversion, which does not depends on the "symmetric and periodic" in any way.
In fact, if you *do* have a symmetric, periodic waveform (the only wave form you can easily phase invert anyway), phase inversion and polarity inversion are one and the same - polarity inversion is just a little broader, not dependent on signal, and super easy to accomplish. The application just has different meaning depending on the usage.