Suppose we take dative case, a common definition given in early German teachings is:
Case used to express direction towards an indirect object, the receiver, and is generally indicated in English by to or for with the objective case.
Would prepositional and adverbial objects as discussed in this post be considered under this definition? If not, why?
They do not, because the rule you quoted refers to dative without a preposition. Usage of dative with a preposition is a different matter.
Also note that this rule is intended as a simple rule for beginners and simple cases, but it does not work equally well for all verbs. The term indirect object does not make much sense in German.
Generally, the term dative object (Dativobjekt in German) refers to dative objects without a preposition, not to objects or adverbials consisting of a preposition and a noun or pronoun in dative. Confusing these is a common mistake and there are countless questions on SE German resulting from this, like this one or this one.
The usage of dative + preposition depends on the preposition and the verb. In the case of an adverbial, the meaning depends almost solely on the preposition. A prepositional object, however, does not have a meaning on its own, so it depends on the verb.
"What is a definition of the dative case then?" - It is a traditional thing to try to explain cases in terms of their "meaning". And there are cases with a meaning, so-called adverbial cases, Hungarian has many of them. German case is different, like all languages with just a few cases. There is a connection to meaning, but it is much more schematic and it is too simplistic to think of a meaning that the dative has in isolation. A responsible way of talking about dative case is to view it in connection of the functions of nominative and accusative, i.e. in connection with the whole clause. In fact it is said that languages only ever have a dative case if they also have accusative and nominative. In the spirit of David Dowty's "proto-roles", the nominative subject is characterised as the participant having relatively more "active" features in a particular clause, and the direct object as having relatively more "passive" features. The dative is a third case, a case for a participant that is neither. Therefore, a recipient is expressed as a dative object because it is not very active and also doesn't change very much either. But the same is true of participants with just the opposite status, those from which something is taken away: "Er nahm es mir weg." Here, "er" is subject because it is most active; "es" is object because it changes most (i.e., changes place), and then there is a third item.
(This is e.g. in a paper by Dieter Wunderlich, Towards a structural typology of verb classes, in: Wunderlich (ed) Advances in the theory of the lexicon, Mouton de Gruyter 2006.)
Concentrate on the verbs. Each verb has its own way of expressing the various "participants", as nominative subject, as accusative or dative (or even genitive, mostly outdated) objects, or as prepositional phrases. (And a dative object has nothing to do with a preposition + dative phrase - it's just a coincidence that it evolved that way.)
Though there are some common patterns that apply to many verbs, this can be very misleading for others. So, better treat "definitions" (like the one you mentioned) as rules of thumb that explain some common cases, but inevitably fail in others.
I'd recommend to learn verbs by complete sentences with subject, objects and prepositional phrases, so your brain has a chance to use its really powerful pattern-recognition capability.
If you encounter a new verb, all the "rules" will not guarantee that you use it correctly. Especially, it's close to impossible to get the prepositional phrases right just from the verb's meaning alone. (For accusative and dative objects, the chances are better, but still far from perfect.)
