Having varying definitions of apposition floated around doesn't mean that this concept is so obscure that it resists definition. Trying to find some middle ground between different definitions of apposition would be a waste of time. The authors of CGEL give a thorough treatment of apposition which includes convincing arguments against analyzing unrelated constructions as appositives. Their definition of apposition is perfectly consistent with the grammatical model they expounded in the book.
Analyzing any syntactic point in detail requires a thorough understanding of the given grammatical model. So if the author decides to include relative clauses, content that-clauses etc. under what they refer to as "apposition", it may work within the grammatical model they developed, if it is done consistently. The problem is that it is often not the case.
I'll try one of the definitions offered here and apply it to the example of "appositive that-clause" provided in the same post:
The term apposition refers to the juxtaposition of two grammatical
units in a sentence with the same or similar referent.
The problem I have with this one is in the use of the term "grammatical units" (which implies that the author of the definition has in view a range of grammatical forms) and its mismatch with the term "referent". "Referent" is defined as "the thing in the world that a word or phrase denotes or stands for." I wonder how we could fit a clause into this definition of "referent" as a "thing in the world". What kind of a referent can the clause "The accused was ignorant of the rules of English law" have in the real world?
[ NP The fact [ that the accused was ignorant of the rules of English law]] will not afford any defence.
A noun phrase, on the other hand, by definition have a referent in the real world - something that we can identify as a sort of an entity being referred to.
The fact is that both that-clauses used as in the sentence above and NP appositives are pretty much fixed positionally, following the noun head, but it doesn't provide enough argument for their identical treatment. On the contrary, doing so obscures the true purpose and function of both that-clauses and NPs as noun dependents.
One difference between the specifying NPs and that-clauses is in their different entailment relation with the head. The specifying NP can, without exception, stand for the NP it modifies. A "that-clause" noun complement, on the other hand, may or may not stand for the NP head, as illustrated in the example from CGEL:
The excuse that he gave, that the train had been late, seemed to satisfy the boss.
Probably the easiest way to understand the difference between the two is by looking into the properties of that-clause noun complements. The striking characteristic of these clauses is that they can be used with a pretty limited number of nouns. They are accepted only with nouns belonging to a specific semantic class. Typical ones are nouns derived from verbs like belief, guess, promise, assumption etc. and adjectives like possibility, confidence etc. This means that noun modification with a that-clause can be compared to the modification of verbs and adjectives by the same clause: He assumed that.. His assumption that.. The probability that.. It is probable that.. etc. This is so unlike noun phrases in the appositive function. The appositive NP is typically a proper name whose head can basically be anything that can be named.
The difference between other formal categories that are lumped under the concept of apposition and the specifying NP modifier of a noun is far more pronounced. In my previous post I said that analyzing an ascriptive NP supplement as a reduced relative clause would require getting rid of the definition of a relative clause as an anaphoric relation between an NP and its antecedent.
Ascriptive NPs, adjective and participial clauses are strikingly different from appositive NP supplements. For one thing, they are very mobile and versatile syntactic constructions. I've discussed their properties at length in another thread, so I'll not repeat the same here.
Finally, what I see as the general idea common to these varying definitions of apposition is the idea of "interruption" or maybe "appendage". It seems that most think of apposition as anything that can be stuffed between commas and that is in some way related to something in the main body of the clause. The problem comes from the fact that any possible grammatical form can be presented as an interruption to the normal flow of the sentence, and literally anything can be enclosed in commas. Of course, we will want to refer to these interruptions by one single term, but this term is very general. CGEL opted for the term "supplement".
The fact is that relative clauses, appositive NPs, content that-clauses, adjectives, nouns and all other syntactic forms can either be detached from or integrated in the sentence structure. They are no more or less related as supplements than they are as integrated in the sentence structure.