I'm a non-native French speaker, but I am pretty confident that's not true. They are actually different sounds, not just the same sound at a different pitch.
French is not a tonal language like Chinese. Pitch is not used to distinguish between different phonemes.
And ê, when pronounced (most of the cases) it's just a è.
ë, contrary as said in the article (full slop?) is the most complicated and with some exceptions. But there is so few words that use that letter that you just don't have to care.
Just pronounce ë as è when its in (inside) a word and not pronounced at all when it's at the end.
The only exception I can think of is canoë (pronounced conoé), but everybody will understand if you say cano.
Ambiguë (ambiguous) and aiguë (acute) [1], but these are "old" spellings.
For instance, this word "ambiguë" was changed in the 1990 spelling reform to "ambigüe" [2] probably to emphasis the fact that the U is not mute (because for most -gue words it is, like for "fatigue" in french and english).
Like with ï and ü, the tréma mark is precisely the mark of an exception.
My first French teacher drew a picture of a smiling triangular-topped tombstone with long eyelashes on the blackboard, the word "acute" written up the left (ascending) side of the top and "grave" down the right hand side. A cute grave. Easy to remember. And fairly useless, since it doesn't help a whit with how to pronounce those accents.
* It's also in their names - aigu and grave, but this requires knowing what these words mean.