> If you want to analyze it that way, you'll find that the semantics are the reverse of what you predict: when you catch fire, it's the fire that takes hold of you.

That is one of the meanings of catch, just like when you catch a cold, the cold takes hold of you, or when you catch your foot on something, that thing took hold of your foot.

> This argument is predicated on forgetting the difference between use and mention. What part of speech would you say an is in that sentence? Is it an article?

Fair enough, though I would still argue that being able to make a noun out of the article in this way relies on them having a stable, recognizable, individual pronunciation.

> This is not a word-level phenomenon in any way; intrusive R also occurs between syllables of a single word, as long as there's an appropriate vowel-vowel sequence.

Well, we are trying to define what a "word" even is, so you can't bring this distinction in. A priori, "saw it" could be a word, just as my whole comment could be a single word. We are trying to come up with a formal definition of what it means to be a word; if we want "an elephant" to be a single word and "saw-r-it" to be two words, we need to come up with a distinction between these that doesn't presuppose that "saw" and "it" are separate words.

> Placing one between saw and it would not normally be viewed as altering the pronunciation of either word (Which one do you think is altered? I guess by nonrhotic standards it would have to be it), but as the application of a general rule.

Depending on the exact accent, not all words follow this rule. In certain accents, at least, it is quite specific to words that have an 'r' in their spelling (well, to words that historically had an r sound that was lost, and is usually preserved in the spelling), so "four o'clock" would get a linking R, but "saw it" would not. So at least in these cases, by your definition, we'd have to say that "four" is not an individual word, phonetically speaking. Also note that linking/intrusive R doesn't appear inside morphemes, normally, only between morphemes, or between morphemes and suffixes. So you get [Kafka-r-esque] in certain accents, but never inside, say, "dais".

> Placing /n/ between a and elephant is not the application of a general rule, it's the application of a rule specific to a.

This could also have been a general rule, that happens to apply to a single word in modern English. Regardless, as I have mentioned before, if you want to define "phonological word" as a unit whose exact pronunciation is only knowable when you have all parts present, then lots of phrases are phonological words in English, unless you add a lot of exceptions to your definition.

> You're saying that people argue over which items count as words, not that they argue over what it means to count as a word.

These are not that different in practice. If you have a good formal definition, then you should be able to say for any object whether it is a word, a part of a word, or a sequence of words. If you can't do that, you don't really have a definition. The definition you gave basically equates word with atomic, but then if "atomic" is not well defined, or even definable, then we're back to square one of not knowing what a word actually means.