A couple of days ago Jim said
> 'Amphitrite', as far as I know, has nver been regarded as one

Certainly not a "big ballad" or "Child ballad", but surely a song that tells a story, and presumably (though I haven't checked) printed on broadsides and so a "broadside ballad".

As so often, and as mentioned above, there are no hard and fast lines. Taking the first few songs in the New Penguin Book:
The Bold Princess Royal: broadside ballad
Bonny Bunch of Roses, O: broadside ballad
Captain Ward and the Rainbow: Child ballad
The Dolphin: broadside ballad
Faithful Sailor Boy: broadside (late): ballad? If one characteristic of ballads is that the stories are pared down to the minimum, this could qualify, but to my mind there is too little story.

Not finding much variety in the first section of the book, so on to the next.

Bold Fisherman: broadside yes. ballad? More concerned with the details of the romantic encounter than with action.
Cupid the Pretty Ploughboy. broadside yes. ballad? Same comment as Faithful Sailor Boy
Cupid's Garden. ditto

I could go on, but I think that's enough examples for now. Most of us would count all of them as folk songs, though Jim would probably exclude Faithful Sailor Boy and possibly some of the others. Some are indisputably ballads, but opinions will differ as to how many of them.

One characteristic that they all certainly share is that "the folk" liked them: a prime criterion for inclusion in the book is that they were widely collected.