The diaeresis and umlaut are literally exactly the same Unicode character and the symbol or diacritic itself can be called either. The difference is what it does to the vowel — it’s a diaeresis if it indicates a repeated vowel, and an umlaut if it modifies the vowel sound.

Just to note, a diaeresis isn't always the same vowel repeated. The most common example is "naïve", and it also occurs in names like Chloë.

Whoops, you’re right, my explanation missed that!