You are all very, very confused.

Umlaut is the modification of a vowel sound, as correctly defined by Webster 1913. This is a linguistic phenomenon found (only?) in Germanic and Celtic languages.

In German, if umlaut takes place in a word, this is indicated by adding an e after the vowel ("ae") or by adding two dots -- a diaeresis ("¨") -- on top, resulting in "ä". English does not follow this convention, and English cases of umlaut show up mainly as vowel changes in Anglo-Saxon plurals, eg. man becoming men, as opposed to German Mann/Männer. Another language with umlaut, Danish, opted to use the æ ligature and ø sign instead.

Now comes the tricky part. The idea of placing a diaeresis up top to distinguish between vowels has been borrowed into many other non-Germanic languages, such as Finnish, whose orthography uses the diaeresis to distinguish between front vowels (äöy) and back vowels (aou). However, these languages do not have umlaut, so all they have borrowed is the notation. The Finnish "ä" is thus "a with diaeresis", not "a with umlaut".

However, you are not alone in your confusion, and the use of the word umlaut to mean diaeresis is pretty firmed entrenched in the English language, as can be seen from the fact that even the HTML character entity for the diaeresis is called ¨. Most modern non-prescriptive dictionaries thus include umlaut as a synonym of diaeresis, but this is still semantically squicky and I applaud Unicode's efforts to set the record straight.

And cheers to Albert Herring, Gritchka and Tiefling for comments and corrections.