Avangi,
As is very easy to do, I think you may be confusing the terms of traditional grammar with the terms of "transformational grammar".
In this case below, for example, you're mixing a basic idea from transformational grammar (the idea of a 'head' word of a
phrase) with an idea from traditional grammar (the idea that a
clause can be named by its function: noun clause, adjective clause, etc.).
Avangi“If we wished to refer to a phrase or clause beginning with an adjective or adverb, I expect we'd say, "This is an adverb clause,"”
An adverb
phrase, a term from transformational grammar, is 'headed by' an adverb. (The adverb (or whatever) need not actually be at the beginning, by the way:
happily is the head of the adverb phrase
very happily.) These are never called 'adverb clauses', as far as I know.
On the other hand, an adverb
clause (or adverb
ial clause), a term from traditional grammar, is a clause (subject and verb) which functions as an adverb (answers questions like
How?,
When?, etc.) A phrase with adverbial function can be a prepositional phrase:
... threw it into the barrel. Here's where traditional grammarians sometimes use the term adverbial (prepositional)
phrase -- and this is where it gets very confusing because that is very close to the term 'adverb phrase' from transformational grammar.

Examples:
Transformational grammar: Adverb phrases
surprisingly, very quickly, as quick as a bunny, as quickly as I possibly coul
d, too late to see itTraditional grammar: Adverb clauses / Adverbial clauses
when I was young, unless we harvest before the rain, although darkness was already upon them______
Notoriously varied, and therefore difficult, is the terminology surrounding participial constructions. The traditional approach typically calls them phrases (because they have no subjects), and the transformational approach calls them clauses (with implicit subjects). Take this example:
Julia accidentally cut her finger opening a can of tomatoes.opening a can of tomatoes is called a participial phrase by the traditionalists. Many of them even insist that it must be placed first:
Opening a can of tomatoes, Julia accidentally cut her finger."It wasn't the finger, but Julia, who was opening the can", goes the argument, and the phrase must be placed as near as possible to
the noun it modifies. (Here you have an implicit claim that this participial phrase is
adjectival.)
On the other hand,
opening a can of tomatoes is called a (non-finite) clause by the transformationalists. The implicit subject is
Julia. They don't insist that it's adjectival, of course, because it answers the questions
When? and
While doing what? and, favoring description over prescription, they are quite happy to leave it at the end of the sentence.
______
To make matters worse, not all traditional grammarians always agree on their own terminology, and not all transformational grammarians always agree on theirs!

I hope this helps!

Jim