| Whereas the umlaut represents a sound shift, the diaeresis indicates a specific vowel letter that is not pronounced as part of a digraph or diphthong.
I.e., graphically there's no difference. Thus I don't really care to call it one word or the other. Sure, 'diaeresis' is more correct that 'umlaut', but most English speakers -I suspect!- are more likely to recognize the latter than the former.
As I said, there is a very distinct difference in writing. The umlaut is written more like ő with two short lines (handwriting only!) whereas the other is written ö (two dots).
It might just be a stylistic thing that distinguishes not diaeresis from umlaut so much as German handwriting from others'. That is, it might be that Germans don't distinguish between the two themselves, and then also the French, the Spanish, etc. If that's the case then I'd say there's zero difference between the two. A test of this would be to ask a German person (or several, including linguists as well as non-linguists) to handwrite a German word that uses umlauts and also a non-German word that uses diaeresis, and if they write it the same then I'd conclude that there is no difference.
And if it's handwriting only, then is there a difference as far as Unicode goes? No, there is not. In Unicode diaeresis and umlaut are the same (U+0308 Combining Diaeresis).
Why are you putting words in my mouth? And when did you say you taught this?
The evidence I'm finding is that there is no semantic difference between umlauts and diaeresis, and that the only real difference is in handwriting style, and I'm then wondering whether that style difference has to do with the language one is writing in or in one's upbringing, which is a very fair question to ask.
The style difference seems smaller and less noticeable than the handwriting difference in Latin ligatures. If the difference lies with what language one is writing, then it is closer to a real difference, but if it lies in where one learned, then it's a negligible difference.