What does the concept of "apposition" mean precisely?

The Cambridge Grammar of the English Language makes a clear distinction between ascriptive and specifying supplements, and categorizes only the former as apposition. I believe that answers to similar questions should refer to the diversity of opinions on what constitutes apposition. For example, the Wikipedia (yes, I know!) article includes the following examples:

The first to arrive at the house, she unlocked the front door.

and

A staunch supporter of democracy, Rob campaigned against the king's authoritarian rule.

Do these examples fall under what is designated as "apposition" among linguists?

All syntactic forms can be integrated in the syntactic structure or detached from it by means of punctuation. There is an "integrated vs supplementive" distinction that cuts across the language. I always think of the integrated sentence elements as more basic than those loosely attached to it. So, there are integrated NP elements and supplementive NP elements, integrated relative clauses and supplementive ones, integrated non-finite clause and supplementive ones etc. Here are some important characteristics of supplementation:

• Regardless of the form, all supplement forms are understood as appendages to the core structure of the sentence.
• Any supplement has an anchor in the main body of the sentence
• Supplements are set off from the rest of the sentence by punctuation
• There isn't one single syntactic or semantic relation between all forms of supplements and their respective anchors.

On the issue of terminology: for a long time, the term "apposition" has been used in linguistics to designate a very specific relation between two nouns. Using the term to mean "anything that you can possibly find enclosed within commas, parroting the previously said in related terms" shows a lack of understanding of the language structure, not only of this particular concept.

Now, for the question from the OP - neither of the introductory phrases in these sentences are appositive modifiers. Actually, the relation of apposition can never be established in this grammatical form: NP, Main clause. Even if the sentence is turned into a possible pattern, like:

Rob, a staunch supporter of democracy, campaigned against the king's authoritarian rule.

The NP "a staunch supporter of democracy" is still not an appositive modifier of "Rob". "A staunch supporter of democracy campaigned against the king's authoritarian rule" doesn't entail that it was "Rob" who campaigned. This fact clearly indicates that the phrase is not intended to specify who Rob is, but rather to ascribe a property to Rob.

To understand apposition it is advisable to start with the more basic, integrated pattern. What is exactly an appositive modifier? It is a unique type of a NOUN (post)modifier. What makes it unique? Firstly, it is a specifying modifier (as opposed to ascriptive ones) which further specifies the underspecified noun phrase. This explains why it is typically a proper noun . Secondly, noun-postmodifiers other than appositive ones are limited to meanings related to age, size..(CGEL 446). Finally, and most importantly, this modifier can take over for the whole NP without affecting the interpretation of the sentence. That said, this would be an example of an integrated appositive modifier:

My cousin Rob campaigned against the king's authoritarian rule.

Relative clauses can also be integrated or supplementive, and, like appositive modifiers, the integrated ones are also noun-postmodifiers. This is why we can think of specifying NPs and relative clauses as related, and can maybe group them together, among the wider group of forms functioning as supplements. Their primary functions are as modifiers within the noun phrase structure.

On the other hand, three other forms (among eleven in total illustrated in CGEL as possible supplement forms ) can be grouped together, based on their syntactic behavior: ascriptive NPs, adjective phrases and participial clauses. As supplements, these forms are to be understood as predicates, or else "subjectless clauses". The anchor for these is an NP in the main clause, functioning as the implied subject in these reduced clauses. Unlike apposition, which has a very specialized and limited use, these three forms of supplements are used very widely, in a variety of ways and positions.

In one of the other answers to the question in the title @herisson says "as far as I can see there is not in fact a unanimous consensus among linguists on the definition of terms like "apposition"/"appositive" etc."

This is indisputable. For example, The Oxford Dictionary of English Grammar (p32) says in its discussion of apposition:

Grammarians vary in how widely they apply the term apposition.

Peters in the section on apposition in The Cambridge Dictionary of English Grammar (p40) states:

The term apposition refers to the juxtaposition of two grammatical units in a sentence with the same or similar referent. They may or may not have exactly the same grammatical structure, and it may or may not be possible to omit one of them without impairing the syntax of the sentence. Thus there are several parameters involved in apposition, on which grammarians diverge.

McCawley in The Syntactic Phenomena of English (p467) goes into more detail:

Various authors have proposed criteria for identifying a combination of two expressions as 'apposition', the most popular ones being (i) that the two expressions have identical reference; (ii) that they be of the same syntactic category; (iii) that either of them can be omitted without affecting the acceptability of the sentence; and (iv) that either of them can be omitted without affecting how the remaining constituents are interpreted.

...

There is little consensus as to the appropriateness of (i-iv) as criteria for 'apposition' or with regard to how to apply the criteria ... .

The index in A Comprehensive Grammar Of The English Language (Quirk et al., p1673) has over 25 sub-topics under the headings apposition and appositives, including attributive, partial, restrictive, nonrestrictive, strict, weak, clause., of-construction and many more.

It is of course legitimate to regard one particular grammar's treatment of the issue as the most persuasive (e.g.The Cambridge Grammar of the English Language's). But it is clear that the question What does the concept of “apposition” mean precisely? does not have a definitive answer.

To conclude, here is a small example of the difference in classification between the The Cambridge Grammar of the English Language and the Oxford Modern English Grammar, which was published ten years later:

In the section Appositives vs complements (p446) the The Cambridge Grammar of the English Language says of the sentence The suggestion that they cheated was quite outrageous:

The content clause (that they cheated) does not qualify as an appositive; it is a complement licensed by the head noun suggestion, just as in a clausal construction.

In the section called Clauses functioning as Complement in noun phrases (p123) the Oxford Modern English Grammar lists a set of nouns that take finite or non-finite clausal complements, for example chance, fact, idea, occasion, proposition, question and states:

Clauses that function as complements of nouns are often called appositive clauses.

The following sentence is included as one of the examples:

[ _NP The fact [ _clausethat the accused was ignorant of the rules of English law]] will not afford any defence.

So, to finally answer the question at the end of the post, the two sentences cited are considered as examples of apposition by the author of the Wikipedia article in which they are listed, but not by The Cambridge Grammar of the English Language.

You may want to ask this or a similar question on Stack Exchange: Linguistics, which is no doubt frequented by more linguists than contribute here.

The answer by Rejlan Givens that says "for a long time, the term 'apposition' has been used in linguistics to designate a very specific relation between two nouns" gives only one of various definitions of "apposition" that have been used in linguistics. It is not the case that all linguists have historically defined apposition in such a way as to exclude predicates/"subjectless clauses".

There may be good reasons to use such a definition, but that isn't the same thing as saying that for a long time, the term "apposition" has only been used by linguists in the specific way that it is used by the authors of the Cambridge Grammar of the English Language (CaGEL). Given that the question post says "answers to similar questions should refer to the diversity of opinions on what constitutes apposition" and asks about "what is designated as 'apposition' among linguists," this question appears to be at least in part about the actual history of the term "apposition", and not just about the best way to use the term in future works, or about the correct analysis of the phenomenon itself.

Unfortunately, I do not have an expert understanding of any alternative definitions of apposition, so I can't say how they compare in terms of rigor or consistency to the CaGEL definition, or whether any sources that use a broader definition have alternative terms that mean the same thing as CaGEL's narrower definition of "apposition". One source that I know of that seems to refer to alternative definitions is "Appositional constructions", by Herman Heringa, 2011. Heringa says that "O’Connor (2008) [...] argues that appositions underlyingly are non-restrictive relatives with a null relative pronoun as its subject" (p. 14) and mentions that Doron, E. (1994) ‘The discourse function of appositives’ argues that "appositions behave as nominal predicates" (p. 76).

Relevant previous ELU posts and comments:

• comment thread between BillJ and Shoe: What is the grammatical term of "whose wife was a school teacher"?

• my answer to It's the size of a brick; What size shirt/shoes do you take?; I have a daughter your age

• my answer to Appositive with no definite article