What are adjective phrases exactly?

I've been under the impression that "adverbial phrases" are phrases that function adverbially and "adjectival phrases" are phrases that function adjectivally. My understanding is PP and AdvP can overlap, and so can PP and AdjP. A prepositional phrase could be an adverbial phrase or adjectival phrase.

But information on Wikipedia is contradictory in this regard. The Wikipedia page for "Adjective phrase" says: (by the way, that page is slightly messy, possibly vandalized)

An adjective phrase (or adjectival phrase) is a phrase the head of which is an adjective

Per this definition, adjective phrases are defined in much a similar way to noun phrases and prepositional phrases. However, another Wikipedia page has this line:

She's [the woman with the hat]. (adjectival phrase, in this case a prepositional phrase, modifying a noun in a noun phrase) (source)

Clearly "with the hat" is a prepositional phrase with no adjective head. So which definition is correct? Can prepositional phrases also be adjectival phrases?

Solution 1:

The confusion stems from calling anything that modifies a noun an adjective, and the only slightly better solution of saying 'adjectival' instead of 'modifier in a noun phrase'.

The Oxford Dictionary of English Grammar p9

adjectival (n. & adj.) Loosely, (a word, phrase, or clause) behaving like an adjective (including single-word adjectives); e.g. in a damp cloth, the word damp is an adjectival element.

The term is also used for examples like the following:

guide price

the greenhouse effect

the man in the white suit

an I’m-all-right-Jack attitude

Some writers informally use the word adjectival to describe all of the italicized strings (or even say that they are adjectives), but this is infelicitous, since form and function are being confused: the first two examples involve nouns as modifiers; the third example involves a prepositional phrase; and the final example has a clause as modifier. Note that the terms adjectival and adverbial are not entirely comparable. Adverbial can denote one of the main elements (1) in clause structure (the others being subject, verb, object, and complement); it is then a function label. Adjectivals operate at a lower level, often as part of a noun phrase (which itself may function as subject or object). An adjectival may in some instances be the sole realization of a complement (e.g. You look hungry), but the adjectival as such is not a functional element in clause structure.

In modern grammar form and function are treated separately. Adjective phrases, noun phrases, prepositional phrases and so on are defined based on their internal structure, and the range of possible functions they can fulfill in larger constructions. Any particular instance of a phrase has a specific function: subject, object, modifier, complement, etc.