I'm not sure whether this is what WHONOES meant, but here's one view of the difference.But the shift is constant from pulse to pulse so show me how that creates distortion.
The first two plots show two cases of a triangle threshold waveform. The first with an instantaneous signal level of 0.0; the second with signal level 0.5.
The purple trace shows the fundamental of the resulting PWM waveform. We assume that the sampling and filtering is good, so all harmonics above fundamental are to be ignored. The change in siganal level is seen to affect the magnitude of the fundamental, as we would expect and does not affect the phase of the fundamental.
The second two plots show the same conditions with a sawtooth. Here the phase of the fundamental can be seen to shift with the signal level, as pointed out, as one end of the PWM output is fixed in time, a regular, uniform sampling time. So the phase of the fundamental changes with signal instantaneous level.
I have not tried to quantify what level of distortion this phase shift would mean for different input waveforms. Seems it could be argued that for adequately over-sampled waveforms (that is no "rapid" level changes) and some applications some phase distortion could be tollerated.
Another aside: I also remember that some implementations tended to add the threshold waveform to the signal and then have a comparator to a constant threshold level. Can't remember whether having a constant threshold into the comparator was significant or for what subtle reason.