Even the name is unfortunate, reflecting its historical origin. [...] Since, unfortunately, in classical mechanics the second charge only comes from rotations, this leads to the mental picture of an electron rotating, however this is false.
So ... let's see if I've got this right. The reason for the name is that
if the world was classical instead of quantum, then the Stern-Gerlach experiment
would have been measuring how the particles spin. The results of this particular experiment were consistent with them actually, classically spinning, but only at a given rate. So people exclaimed "Ah, spin is quantized!" But really they weren't spinning at all, as has been shown by other experiments (which you allude to and I know nothing about.) It's just that their magnetic properties, not actually caused by them spinning, gave results
consistent with the hypothesis that (a) they were spinning (b) the spin was quantized.