> Ë with diaeresis is the easiest case to deal with. The diaeresis (the two dots) signifies that the underlying “e” is pronounced as /ɛ/ (as “e” in “bet”, i.e. the open e), no matter what comes around it, and is used in groups of vowels that would otherwise be pronounced differently.

Yes, but there are other uses. For instance, in "ambiguë", the ë itself is silent but signals that the u before it is pronounced as a standard u. Without the diaeresis, the u itself would be silent but would make the g hard (in French, g before e is soft).