1. older than the 1960s.

Along with Yarmouth Town the song is credited to Peter Bullen's grandfather. Both were sufficiently convincing to no lesser a personage than Peter Bellamy to include on his Mainly Norfolk album of 1968. Would he have done this is if they weren't 'genuine'? As had already been suggested, if they weren't then Bellamy would have been the first to have admitted to it.

2. has ever circulated in a form very different from the established text and tune

This could write off many Broadside and other ballads. I've already mentioned (and linked to) the equally hackneyed Broadside of Out With my Gun in the Morning which we find unchanged in the singing of Jimmy Knights (VOTP 18) who said elsewhere he got it from a written source. Would we find such lines as back again in the sunset's glow in a Traditional Song? Of course not, but this doesn't devalue its essense - on the contrary, like the language of Fakenham Fair it serves to make it still more genuine as a piece of vernacular songwriting, someone over reaching themselves in terms of image, but such ambition can only ever be a good thing. As I keep saying, whatever the shortcomings of the poetry, it sings like a dream.   

3. was even moderately well known in the rural or urban working-class community that it's supposed to represent.

We might bring innumerable other songs that weren't in the least bit well known or of interest to working class communities - the North-East in particular is celebrated for this sort of thing. I wouldn't say the songs of Tommy Armstrong, for example, are especially well known, however as a Balladeer well versed with the idiom of Traditional Song and Music I have no hesitation in thinking of his songs as being Folk Songs. More obvious Fakesongs (The Blackleg Miner) I have less sympathy for, though the various reconstructions of Captain Bover confirm a notion that such art is not entirely lost to us.

4. embodies the attitudes and ambiance of that community, or any community other than that of sophisticated singer-songwriters

Like the writing of innumerable Broadside hacks, in no way shape or form can we think of Fakenham Fair as the product of a sophisticated songwriter. It is Folk Art, pure and simple - composed naively in an idiom which is as much the traditional cause of the song as the experiences the maker of the song wishes to share with us. That said, there's no reason makers of Folk Songs can't be considered sophisticated songwriters - Tommy Armstrong was one such, George Bruce Thompson was another and reading through my Faber Book of Popular Verse the evidence is right there for countless others. In terms of sophistication and self-consciousness one has Fakenham Fair on the one hand, and The Hiring Fair on the other. Contrast and compare!

4. was ever known to anybody other than folk revivalists, revival aficionados, and (presumably) their non-revivalist source

Again, I think we must trust Bellamy on this one, and the fact that both Yarmouth Town and Fakenham Fair are namechecked in Dick Bagnall-Oakley's celebrated introduction on Bellamy's Won't You Go My Way, recorded live with Louis Killen in 1971. You may download the intro & first track gratis from HERE for the next few days. I've collected tunes, songs & stories from non-revivalist sources which I've every reason to believe were the work of the singer I got them from - in my experience such people are very modest, they make thiese things as they make bread or cultivate their allotments, by way of a more innocent mediumistic creation.

Of course, if consistency and reason don't matter, one may define "folksong" however one pleases.

There are many 'folk songs' which are the work of known individuals - check out M'Gintie's Meal and Ale, written by George Bruce Thompson and immediately included in the Grieg and Duncan collection, and the works of Tommy Armstrong. These songs are made in the Idiom of Folk and Traditional song by individuals familiar enough to work with it as a creative medium. One could say the same of Broadside hacks & Gallows balladeers, however so artless their efforts, they still ring true enough as a trawl through the Axon Ballad archive will reveal. Do we dismiss such classics of Folk Song Idiom as The Kielder Hunt and Til the Kye Comes Hame because we know who wrote them? I think not. Our understanding of the idiom of folk song must never be tied to romantic clauses (however so attractive they might be) that preclude such obvious exceptions. Folk Songs are formed by the Traditional Idiom, but whether or not they then enter that tradition and become traditional themselves is pretty much by the by really. I can think of many that didn't in collections as diverse as The Child Ballads and the Copper Family Songbook, but that doesn't disqualify them as being Folk Song, anymore we may disqualify Fakenham Fair.