I always thought accusative case is used for the direct objects of a verb, until a few minutes ago when I found in German grammar in a nutshell, that
Can you please explain the subtle difference between the usage of direct object and accusative case with example?
Direct object and accusative object are nearly always the same thing in German, however there are a few exceptions. Consider a verb such as lehren 'teach' or kosten 'cost' that in the standard language take two accusative objects (although colloquially one often hears the more regular dative-accusative alignment pattern with lehren). Only one of these is a direct object.
Consider a sentence such as Ich habe die Studenten komplizierte Mathematik gelehrt 'I taught the students complicated mathematics'. The direct object here is komplizierte Mathematik, as a simple passivisation test will show:
In diesem Raum wird komplizierte Mathematik gelehrt 'In this room, complicated mathematics is taught' is a fine sentence in German, whereas In diesem Raum werden die Studenten gelehrt is not. So, while both objects are in accusative case, only Mathematik is the direct object.
You have to be careful with the passivisation test in one respect. There may be semantic reasons why a verb cannot be passivised. This is particularly true of verbs where there is nobody doing anything voluntarily, such as treffen in the sense of 'to come across someone' (neither hitting someone with an object nor the reflexive use 'sich mit jmdm. treffen' is intended here).
Ich habe sie im Bus getroffen 'I met her in the bus.' can for most people not be passivised to Sie wurde von mir im Bus getroffen. The reason lies in the semantics of treffen where none of the participants actually acts and does something, which often bars passive formation. This has nothing to do with the objecthood of sie.
For this reason Dies kostete den Mann 50 Euro. 'This cost the man 50 Euro.' cannot be passivised to 50 Euro wurden den Mann gekostet or Der Mann wurde 50 Euro gekostet. But with German requiring the indirect object to precede the direct object in unmarked Subject-Verb-Object-Object sentences, we can safely assume that den Mann is an indirect accusative object, because Dies kostete 50 Euro den Mann. sounds really strange, at least without an appropriate context.
Other cases where one could argue whether the accusative object actually is a direct object would be sentences such as Mich träumte, ein Ungeheuer fräße mich. 'I dreamt that a monster would eat me' and Mich dünkt dies gut. 'I think this is good.' However, there are many opinions on what the accusative argument is in these cases (some linguists even arguing for accusative subjects) and such constructions are in many instances on the way out of the language (the two examples I just provided sound really dated), leading to regularisation of the verbal argument marking. (As a funfact, English has remnants of these things in methinks and meseems.)
Furthermore, notice that the definition in the picture that you provided wrote about "some languages". In Icelandic, for instance, there is way more of this "argument marking irregularity" than there is in German.
It's quite simple. When talking about German grammar, don't use the term direct object — and indirect object for that matter.
They simply don't apply. German grammar concerns about grammatical cases. Accusative and dative are the ones that kind of correspond to direct and indirect objects.
The thing is that most times the accusative is the direct object and dative is the indirect object. Most times, but not always.
An example where it doesn't match is English "I thank you" and German "Ich danke dir".
So, to repeat the most important takeaway: Forget the terms direct and indirect objects when talking about German grammar.
An interesting question. The German wikipedia says that an "indirect object is essentially congruent with the dative object. This is certainly the case with normal transitive verbs which govern both an accusative and a dative object. The article also has a short comparison with English objects and notes that the English terminology is different because the object's case is hard to diagnose, due to a lack of noun endings.
But we are interested in the non-fitting cases. The rather rare verbs which govern exclusively a dative object are the opportunities for exceptions.
This page has a more detailed discussion. One explanation is that dative-only verbs implicitly carry a virtual "direct" accusative object, as in "Ich helfe ihm": I bring him (D) help (A). But there are a few cases where that is hard to uphold ("Ich begegne ihm.").
There are also the even rarer verbs which exclusively govern genitive objects: "Er bedarf des Weines." (He needs wine.) "Er gedenkt seines Onkels." (He commemorates his uncle.)
My gut feeling is that these have the function of direct objects in English, so that yes, in the normal case accusative objects are the direct objects in German, but that at least some verbs which don't govern an accusative at all have direct objects in other cases.
