> The Latin-derived cases
They're not Latin-derived, they come originally from Proto-Indo-European (which had even more cases). Many other Indo-European languages retain cases (Slavic languages, Greek, etc.), but were lost in English and the Romance languages.
What does come from Latin is the way we name and analyse these cases traditionally.
Interestingly, several different ways of analysis are in use. I first learned German as a third language and then moved to a German-speaking country, and realized that the way German-speaking people think about their grammar is often different from how foreigners think about the German grammar. The rules end up with the same result, but the angle can often be different.
For instance, we first learned what a direct object was (something which is done with/to, e.g. in “I ate the ice cream”, “the ice cream” would be direct object). Then we learned that in German, the direct object is declined in accusative (which primarily affects the article, and adjective declination). This was consistent across multiple classes and teachers and books and schools. But my German German teachers had never heard of the concept “direct object”; for them, only the “accusative object” existed. Of course, the accusative object would be in accusative, but also, its presence would signal e.g. whether to use “haben” or “ist” for “is” in certain situations (for which I learned an entirely different set of rules that they had never heard of).
You would think that this is because my native language (Norwegian) has different concepts, but our entire way of teaching Norwegian grammar was uprooted at some point pre-WW2 _precisely to map well to German_, to prepare students for German classes when that was a more common second language than English was. (There were tons of things I never understood why were important until I got to apply them to German later.) So you'd think they'd match better.
We learn direct and indirect object in school as well it’s just not what people remember because they either had to grind the Latin for a test or the number (which js stupid on many different levels).
To be technical, "accusative" etc. are cases (i.e. forms of words) while "direct/indirect object" are grammatical roles - those are different categories. In German, for example, the Dative case can mark an indirect object (although some verbs may require the Genitive case for its indirect object), but it can also have other functions. This is even more pronounced in e.g. Latin where the different cases can have a wide range of different functions, not just direct/indirect object.
This is possibly not something that is taught very explicitly in school, but it's what the terminology means. (Or at least it's how I was taught. Linguistics being such an old discipline used to analyse so many different languages means that different people will use terminology differently.)
I know. But like I said, my German German teachers (all three of them, IIRC) used “Akkusativ Objekt”.