But what the two dots are not: they are definitely not diaeresis.
You don't think of "y" being "u" with an Umlaut. For you this is a completely seperate letter. Ditto with ÄÖÅ im Swedish and ÆØÅ in Danish (Æ is never thought of as combined A and E).
No. The German alphabet contains 26 letters, A-Z, plus 4 special letters, ÄÖÜẞ.
The Swedish alphabet contains 29 letters, A-ZÅÄÖ.
Otherwise, except when discussing German, I think "two dots" is much more common: https://en.wikipedia.org/wiki/Two_dots_(diacritic)
In English, ¨ is a diacritic mark called diaeresis, which indicates that a vowel is distinct, and shouldn't be diphthongized or dropped: Coöperation, Noël, Brontë.
In German, ¨ is a diacritic mark called umlaut, which transforms the vowels A, O, and U into their umlauted versions: Ä, Ö, and Ü. These characters are not distinct letters of the German alphabet, but belong to a special weird in-between class.
In Swedish, ¨ is not a diacritic mark, does not have a name, and is simply an integral part of the letters Ä and Ö, which together with Å are distinct letters of the Swedish alphabet. The dots aren't modifiers, they're not optional, Ä is not sort-of-an-A, it's as distinct from A as any other vowel, and its pronunciation is closer to E than A.
Ä in Swedish originated as the AE ligature where the E moved upwards and above the A until it became stylized as two dots. In Danish and Norwegian, they instead promoted Æ to a distinct letter.
Wikipedia does a great job for example for Ü, distinguishing between U-umlaut, U-diaeresis, and letter-Ü: https://en.wikipedia.org/wiki/%C3%9C
Well you've just nerd-sniped me. I can't resist adding this bit of hyper-specific knowledge: it is called a tittle.
Conceptually umlaut dots are different from trema dots, but who cares?
Unicode also makes a difference, but generally recommends the merged character.
This is incorrect, from a linguistic perspective, but it's absolutely the case that if I asked 10 of my friends to describe Ä, 9 of them would say "'A' with an umlaut".
The vernacular not matching expert terms of art never ceases to annoy people, but it's not really worth fighting, I don't think.
Absolutely.
Yet it's hardly a secret either and something you would expect someone who graduated university writing for a relatively well known newspaper to be able to get right. It's trivial to most readers of English whether they got it right, sure. It's also trivial to get it right and they did not. It looks bad and casts bad light on the newspaper for accuracy and getting details correct. Needless to say there are plenty of details that may not be as trivial to get right but are non-trivial in their effect of your understanding of the story. How does something as batshit simple as this winding up smeared on the WSJ's face in such a nothing story affect the readers' estimate of how often they get important details wrong? That assessment is up to each reader but I'm pretty sure we can all agree it probably doesn't help much.
Anyway, the very first paragraph has this:
ÄLMHULT, Sweden—Amid IKEA’s colorful staged living rooms, piles of umlaut-laden housewares and endless rows of flat-pack boxes is furniture that can have shoppers wondering: How does that chair cost only $35?
Which I think is really bad writing for a publication like the WSJ. Swedish does not have "umlauts". We have three letters more than the US English alphabet, but only two of them (Ä and Ö) have the double dots: the third has a ring (Å). These three are real first-class letters though, they are not "umlauted" (which means changed from some base letter due to grammar).