We all seem to be able to agree that "traditional" music is "folk".

I don't. All music is Traditional, but not all music is Folk. All Folk music is Traditional though.

We may find that hard to define

It's easy as ABC; elementary musicology in fact.

but I don't think that's as important as is made out

It's an ongoing bugbear for sure, but not without good reason.

I can't define an elephant, but I know one when I see it.

That's because elephants exist and aren't some rare breed of folk species that only comes into existence by subjecting it to an absurd & unrealistic criteria to prove it is biologically different from any other breed of animal.

It seems to be the people who want other styles of music also to be called "folk" who complain most loudly about "purists" trying to keep them out

Who does this? Or maybe you think by pointing out the Myriad of Styles you can call Folk these days by way of their context - i.e the many sorts of music you hear in Folk Clubs or discussed on Mudcat - this opens the floodgates? BUT it's not a matter of wanting this, far from it, rather it's a matter of facing facts. Nothing to do with Horses either, just to do with sorts of things your likely to hear done in the Name O' Folk these days.

but I have to question their motives for wanting to be included.

There aren't any motives; it's all down to the Come All Ye inclusivity which is the nature of the Folk beast - be it bands, clubs, festivals, Mudcat or whatever. Where there is Folk there is always going to be all sorts of music and approaces that will irk the self-styled Purist, not because the music is at fault or doesn't belong, but because Purists are complete and utter twats who know nowt about the music they claim to love. If they did, they wouldn't be purists. What they do know about is a perverse small minded craving for Rules and Regulations so they may adopt a Jobsworth attitude to further the misery that their anal view of the world dictates they must share.

I've met a few, not many as I say, for which I am truly thankful.