You seem to be just about the only person who thinks that. True, we have some difficulties in defining the boundaries but the same can be said for most musical genres.

Give me a single example of Folk Music as an exclusive corporeal genre & I guarantee you a pint if ever our paths should cross.

I'm now confused by your introduction of the term "Traditional Balladry". Are you now excluding traditional songs which are not ballads? Or is this just a fancy way of referring to traditional songs?

I used the term Traditional Balladry (implying the stuff I do personally) because it can be defined as a genre, whereas Traditional Music can't (What sort of Traditional Music? Irish? Norwegian? Northumbrian? Sephardic? Inuit?). Traditional Songs might include ballads too, of course. As I say, even the 1954 Definition isn't a defining Folk Music in terms of a genre.

I can see differences in the styles in which the two versions of "Saints" are performed, but they are fundamentally the same song

They are conceptually the same song certainly, but the corporeal, cultural & contextual realisation makes them very different. It is not merely a matter of style.

and the structure of the tune says "jazz" rather than "folk".

That is to treat Jazz and Folk in terms of being genres, which they aren't, although Jazz would appear to be the more exclusive of the two in terms of musical parameters. There is, for example, a considerable difference between Folk Rock and Jazz Rock, not just in terms of genre, but of pragmatic usage. Rather like Pork Butcher and Family Butcher.   

I understand the point you're making, but it describes the performance rather than the song.

A Eureka moment here methinks. The performance is the context, the point where the conceptual becomes corporeal; when a mere idea is brought forth into the world to shine like a star, which both of these do, for me anyway. The difference is that whilst you'd really have to have your shit together to play with Louis Armstrong, such a consideration is less important with the late Matt Armour, in whose company even the most timid shaky-egg player would have been welcomed into the fold. As I say I love them both - I never knew Louis personally, but I knew Matt, whose musical parameters were as big as his heart.

If I only listen to the soundtracks rather than watch the videos, I've no way of knowing that one took place in what you call a "folk context" and therefore no way of knowing that the song has suddenly, if temporarily, undergone genre reassignment.

Oh no? Put a blindfold on and I'll see if I can come up with some more examples! But seriously, with folk music it is really the being there, the living breathing inclusivity that demands we check in our egos, aspirations & expectations at the door, which I'll be doing tonight, in The Steamer in Fleetwood, where anything can happen & probably will. Such is Folk Music.

*

Meanwhile, an almost relevant anecdote.

In the good old days in England Sam Smith's pubs carried music licenses and sold cheap (though barely drinkable) bitter such as Old Brewery, which at one Durham public house could be had for a quid a pint, thus making it very popular with musicians. Thursdays was the Folk Club; Tuesdays the Trad Jazz, and Mondays was the Irish Session, the players of which took themselves Very Seriously Indeed, and rightly so in terms of the impeccable standard of their playing which existed in direct correlation to the utter tedium it inspired in the casual listener, such as myself. Said public house was also the scene of many an hearty outrage; one night, for example, I was in there when a fight broke out in the bar between several inebriated men of the same family after a funeral. It was a Tuesday, and the Dixieland Jazzers played on as the chairs flew, and the men brawled, and the locals stood there as if nothing was happening. A lovely summer night it was as I recall, the sun shining, the doors open, and everything at peace with the world; a peace barely disturbed by the proceedings in the bar.

Anyhoo. One Monday night after an arduous coach journey from London I popped in for a pint (those who say to travel hopefully is a better thing than to arrive have never travelled by National Express). The Session Musicians were through in the club room, playing their particular brand of music with fierce concentration and earnestness - a music which filtered through to the bar as a mildly irritating ambience: difficult to ignore, but not really loud enough to engage your attention, especially when one was in there on one's own, enjoying a solitary pint of an autumn evening with a half-ounce of Golden Virginia (Job papers & Swan Vesta matches) and a copy of Heart of Darkness (if only to get a literary measure of Apocalypse Now). Into the bar comes an old lady in her slippers, hair-net and dressing gown. In the absence of the barmaid, she helps herself to a large glass of Grouse from the appropriate optic. Taking a sip, she savours the poison, pondering all the while the nature of the entertainment taking place through in the club room, where our Session friends are playing with such indefatigable gusto they might well get through the whole of O'Neill's before closing time. Then a look of realisation dawns on her wrinkled face as it all becomes clear; something at least approaching a smile plays about her lips as she turns to me (there is, alas, no one else in the bar) and utters the immortal words:
"Eh, that's that Riverdance music isn't it?"
"It most certainly is," I reply, happy for the first time since parting from my girlfriend at Worth Abbey some ten hours earlier.