Dear Faldage. Sorry about retarded peristalsis quip.
I think it an error to say the umlaut is a form of dieresis.
In German script, that is handwriting, a small "e" is two vertical marks. As far as I know, it was just a sort of time saving convenience to put a miniature "e" over a vowel, instead of using two normal sized vowels.
I never thought to ask the prof why Goethe didn't use an umlaut in his name. Now that I think of ir neither did Hitler's propagandist Goebbels. But his Luftwaffe guy, Gõring, did.