I know what an "absolute phrase" is, but I just like to know why it is called "absolute". Which feature the adjective "absolute" refers to exactly?
Here and some other sources do not explain the name.
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Test Magic at your link has a definition, which is worth quoting at some length:
Definition and rules. An absolute phrase is a modifier (quite often a participle), or a modifier and a few other words, that attaches to a sentence or a noun, with no conjunction. An absolute phrase cannot contain a finite verb.
Absolute phrases usually consist of a noun and a modifier that modifies this noun, not another noun in the sentence.
Absolute phrases are optional in sentences, i.e., they can be removed without damaging the grammatical integrity of the sentence.
The reason they are called absolute is because that word is derived from Latin. OED has
Etymology: < classical Latin absolūtus complete, finished, perfect, pure, unqualified, unconditional, unambiguous, (in grammar) in the positive degree, in post-classical Latin also independent, free, unrestricted (6th cent.), (of space or time) independent of any observer and any phenomena...
That is, an absolute phrase is one which is complete and not dependent on another part of the sentence — which is why it can be detached from the sentence without "destroying its grammatical integrity". It could just as easily be called independent, but independent and dependent clauses do contain a verb: some other term was needed, and grammarians like their Latin.
See OED again. The grammatical meaning comes first.
A. adj. (and adv.)
I. Free from dependency, autonomous; not relative.
a. Of a clause, construction, case, etc.: not syntactically dependent on another part of the sentence. Of a word: used without a (customary) syntactic dependant; spec. (a) (of a transitive verb) used without an expressed object; (b) (of an adjective or possessive pronoun) used alone without a modified noun.