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.
I’m speechless.