context

I understand there are different theories of grammar.There is the a set of traditional pragmatic grammars aiming at teaching languages, which might not even have names for themselves.Then,there is a set of grammars for academic purpose, two popular subsets are the dependency grammars and the phrase structure grammars.

Whatever the building principles of a certain phrase is depends on the specific grammar it refers to, though there is a common layman understanding of it simply being a few words with a concrete function in a sentence.

main issue

I met different definitions of what adverb phrases/adverbial phrases are.

Some state, they are nothing more but adverbs enhanced by one or more further adverbs

very suspiciously quiet

other say they are prepositional phrases functioning as adverbs

I threw a stone into the bucket

and still others say they consist of an adverb enhanced by other adverbs and prepositional phrases.

He wrote tremendously lively for a mummy of 4000 years

I named just these , but I stumbled upon more , giving me the impression of arbitrarity.

Please shed some light, for instance by explaining how the adverb or adverbial phrase evolved, which term emerged first, when did the semantic explosion occur or however you think would make sense.