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.