Since German is a language that loves compound nouns, we can use those to distinguish between Umlautzeichen, Umlautpunkten, Umlautbuchstaben, Umlautphonemen and Umlaut (as a historical sound change as well as a present-day alternation of stems).
1. While the English term umlaut can refer to the dots, the German term Umlaut does not. If you need to refer to the dots, you can use Umlautzeichen (marks) or (Umlaut)punkte (dots). Note that Umlautzeichen is more generic: while today, only ü is used, historically there also existed spellings such as uͤ, uͥ, ú, with different kinds of diacritics used as umlaut marks.
2. The letters ä, ö, ü are Umlautbuchstaben. Each of these letters stands for two different sounds, a long and a short vowel: for instance, für has [yː], fünf has [ʏ]. Note that for neither word there is a relationship to a word with u; which means that ü is not some sort of modified u, it's a fully independent letter used to represent one of two specific phonemes.
The second part of Umlaut, Laut, signifies (speech) sound. The speech sounds known as Umlaute, Umlautphoneme, are: [yː] für, [ʏ] fünf, [øː] schön, [œ] zwölf, [ɛː] spät, [ɛ] Hälfte, [oʏ̯] häufig .
3. In both English and German, umlaut stands for a sound change (fronting or raising) of the type: man, men; foot, feet; mouse, mice; Mann, Männer; Fuß, Füße; Maus, Mäuse. Historically, it was this sound change that first gave rise to new sounds, which then needed to be represented in writing in some way. The sound change itself has been gone for almost a thousand years; the sounds, and ways of writing them, live on. Also, due to this historic change, present-day stem alternations such as Mann – Männ- (as in Männer, männlich) are also referred to as Umlaut (Männer, Männchen, männlich, (fach)männisch have or show umlaut, are umlauted).
4. Note that, despite the Unicode standard not reflecting this distinction, umlaut dots are not trema: Noösphäre has trema as diaeresis marker on ö and umlaut dots as umlaut marker on ä. The Unicode FAQ suggests a workaround in case such a distinction needs to be made (e.g. for purposes of sorting).