Regarding dieresis vs. diaeresis, either is correct but the former is considered a variant.

The symbol itself is used, in the examples, to indicate that the vowel is pronounced as a separate syllable, as in naïve or Brontë. The term umlaut refers to the linguistic process wherein a back vowel, such as a, o or u is dragged kicking and screaming to the front of the mouth in anticipation of a front vowel lurking there in the next syllable. In German this process is indicated by placing a dieresis over the back vowel.

Aintcha glad ya ast?