prepositional phrases - Wikipedia's definition of "Adverbial"

On the Wikipedia page for Adverbials, it says [emphasis mine]

In grammar an adverbial is a word (an adverb) or a group of words (an adverbial phrase or an adverbial clause) that modifies or tells us something about the sentence or the verb.

Further down the page it says that “in the water” is an adverbial complement in:

John put the flowers in the water
_^Wikipedia

But in the sentence above, “in the water” is not a modifier of the verb put. It is a complement and a prepositional phrase because it begins with the preposition in. Why is it called an adverbial? If we remove the complement we are left with:

*John put the flowers.

This sentence is clearly incomplete. If we use an adverb instead of in the water we have:

John put the flowers here.

The sentence makes sense, here is an adverb, and it tells the listener “where” the flowers are put. According to Wikipedia an adverbial is either an adverb or a group of words, so is here also an adverbial? Considering its position within the sentence, does here also function as a complement? If I add further information is it an adverbial phrase, or something else?

John put the flowers here in this vase.

The subject of the clause is John, the verb is put (past simple), the flowers is the direct object, and “here in this vase” is: What?

Questions

1. Why does Wikipedia say in the water is an adverbial?


2. Is “here” also acting as a(n) (adverbial) complement?


3. Is in this vase the modifier of the object (the flowers) or the verb? How do I tell?


4. What effect does the adverb, here, have on the sentence?


5. Is here in this vase an adverbial phrase, or an adverbial complement. Is it called something else?

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_{This post was adapted and inspired by an edit (now deleted) in the question: Adverbial phrase. For those who wish to see the original "edit" see the meta post here.}