Technically the word umlaut refers to the sound change in a front vowel such as a, o or u when the speaker is anticipating a following back vowel. Diæresis refers to the mark itself for whatever purpose it is used.
The history of the diæresis in German orthography is an interesting one. Before I get myself into trouble, perhaps I should amend that to say "the history...as I understand it." The German (individual) doesn't speak of a diæresis at all, but of a distinct vowel. To our a, o, u he pairs ae, oe, ue, respectively, where "e" is the "back vowel" represented by the diæresis. In fact, the back vowel was originally represented in print with a distinct character, "e." Perhaps for reasons of economy, as time went by, the e was joined to the vowel preceding it in a ligature, (as are a and e in "diæresis"). The e then became smaller and migrated to the top of the preceding vowel, facing upwards. This, in turn, was abbreviated to something resembling a tilde (~), but drawn with flourish--I believe the 'cups' each contained a dot. Finally the tilde-shape was dropped and all that remained were the dots.
On the matter of the three additional vowels, in fact they are, of course, diphthongs, and German has no more than the five vowels available to any language. It is therefore tempting to dismiss them as vowels, altogether. It is interesting to note, however, (and I speak as a singer, not as a linguist) that speakers of American English, at least, also refer to diphthongs as vowels. The common pronunciation of the series "a e i o u" is something like this: a=ei; e=i; i=ai; o=ou; u=ua (where the 'a' is very short). So that you might say-with a wink-that have only one vowel, and we misspell it.
