Danish dictionary to weed out gender stereotypes – ‘career women’ are now paired with ‘career men’ and manslaughter is a linguistic offence

This is the same as most words in English with the same equivalent. -man is a gender neutral suffix in most of its use cases, it doesn’t literally mean “Male person”, the same as in “Mankind”.

This is true etymologically (the shared root word of English "man" and Danish "mand" originally just meant "human"), but it would be silly to claim that this is the case for most of its use cases in modern English. Modern English speakers are likely to reanalyze many instances like this as referring specifically to male people as opposed to any person. Heck, part of the reason "man" evolved into meaning specifically a male person is possibly because of an assumption of men as the default actors in society. This is why in English the trend has been to replace terms like "Fireman" with "Firefighter" and "Policeman" with "Police officer", since these don't have the same obvious connection to a word that refers solely to males in modern English.

The movement to add new gender-specific words seems contrary to the movement to remove gendering from words.

These often tend to be different movements. Different language communities tend to have different priorities and methods when it comes to socially-inspired language change, and the trend in English to remove gendering from words is not necessarily the trend in other language communities. In German, pretty much any word referring to a person's role or profession is gendered, and the progressive option is usually to explicitly include both gendered versions in contexts where it's intended to be inclusive. Since most Germanic language still have grammatical gender (English being the exception there), it's not as trivial to make words gender-neutral as it is in English.

This can be frustrating (I'm nonbinary and that's just harder than in English when you have to pick between masculine and feminine versions of words so much more), but it's not necessarily contrary to the movement to inteoduce more gender-neutral language in English -- they're two different attempts to solve the same problem of patriarchal language. It's likely that both strategies are worth using depending on the context.

Interestingly, some languages are actually defined by some official authority, or at least the language as defined by that authority is recognised by the government as being “officially correct”. This is the case with French: the Académie Française prescribes the French dictionary and rules of grammar.

It looks like this is sort of the case with Danish. The Danish Language Council, referenced in the article, is responsible for the official Danish dictionary, “which all government institutions and schools are obliged by law to follow”. So Danish, when used in those institutions, is actually prescribed.

So these languages work a bit differently to English, for example, where as you say the language is purely defined by its use, and dictionaries only exist to describe the language as it is used by its speakers.