Of course I know that mathematically they are equivalent. I know that if you invert a pure sine wave or if you shift it by an odd multiple of pi it looks the same. Duh. But phase shifting only works for the special case of a pure sine wave while inverting the waveform works for the general case. Any waveform, any frequency or combination of frequencies. The math as you present it is a simplistic model that predicts what the waveforms look like, and it breaks down if you feed it anything more complex than a simple sine wave. Yeah, I know, all you want to talk about is a simple sine wave and for that restricted case it works. That's OK by me. No skin off my nose.
But I don't think that there is any doubt in anyone's mind that what is physically happening in our all too familiar center tapped transformer is an inversion, not a time based phase shift (i.e., delay). I use this same type of transformer (though much smaller, of course) all the time in my audio work to generate a balanced (complementary) signal from an unbalanced one. If you look at those waveforms you will see that no amount of phase shift will produce the complement of a complex waveform. Still, in audio many folks who should know better erroneously refer to an inversion of an audio signal as generating a signal which is "180 degrees out of phase". Whatcha gonna do? We all know what it means even if it is technically incorrect. But I digress...
I know that mathematically it makes no difference whatsoever whether you use -sin(wt) or sin(wt+PI) (or sin(wt+3PI) or sin(wt+(10^31+1)PI), for that matter) to represent the complementary waveform in the case of a pure sine wave, but -sin(wt) describes more accurately what is physically happening in the transformer.
I know of no definition of phase shift that stipulates a time delay. Anyway, we are discussing ideal transformers and ideal waveforms. There are no complex waveforms, one cannot tell the difference between a delay of PI and an inversion. There is no noise. To add fine points such as this to the discussion is just a side issue which has no real bearing on the question.
Plus the term is commonly used interchangeably with phase difference.
Perhaps that subtlety is important in audio, but this is not audio.
And, that raises the question: Which voltage is the inverse? Is it V1n? Or is it Vbn?