Jeremy Wheeler and Rabbi Zev-Hayyim Feyer have so far caught what they consider an oversight in our recent Good Word sprachgefuhl. In that writeup I claim that this word is sometimes spelled with an umlaut over the U, which is to say, Sprachgefühl, and explained umlaut as a synonym of dieresis. Although the two are generally used interchangeably, there is reason to maintain a distinction based not on the two dots themselves, but how they are used.
Dieresis comes from the Greek word for “split” and, before English began borrowing words from Modern German, it was used only to refer to an umlaut placed over the second of two successive vowels to indicate that both are pronounced, as in the case of naïve Chloë, Noël, Aïda.
Of course, this alternate spelling is now rare in English and other diacritics serve the same or a very similar purpose. In fiancé, attaché, cliché, communiqué it is the acute that tells us the vowel is pronounced. So the plot, as plots are wont to do, thickens.
I assume the people who named the diacritics did the best they could with what they had to work with. Since we have no word for German Umlaut in English, I still think it reasonable if not preferable to use dieresis for the two dots, regardless of their function.
We do not distinguish between other diacritics on the basis of their use, so far as I recall. Why make an exception here? My use of dieresis followed modern trend of referring only to the two dots placed over some vowels for whatever reason.