if you did a pinch harmonic on (say) an E note, does it ring a different note or it is still an E but at a higher octave? if so ... how many octaves higher?
I think it depends on where your hand is positioned when you hit the pinch harmonic. Different positions will provide different harmonics, just like some won't produce one at all.
say you hit a pinch harmonic on the B string at the 5th fret.
It also depends where your right hand is.
I'm not that consistent in where I hit pinch harmonics, and they sound different.
Try going up and down doing pinch harmonics on your right hand, and you'll see what I mean.
It's all there in the section headed "nodes".