The Howden Hill 'version' turns out just to be the Watersons re-write, uncredited. As mentioned in the earlier discussion, they recorded an arrangement of the set from Kidson's Traditional Tunes, shortened and altered in places, with a chorus of their own invention substituted for the original refrain. In fact, the location remains Howden itself, Howden Hill being mentioned only in the final verse as altered by the Watersons. The Lolley/Kidson text is:

The horsemen and the footmen they all drew nigh,
Thinking that the white hare was going for to die;
She slipt out of the bush and thought to run away,
But cruel were the beagles that caused her to stay

The text of the Watersons arrangement, as transcribed from a recording, is given (see earlier links) as follows; though I suspect that it isn't entirely accurate:

Oh both horse and foot they did them unto the place draw nigh
Thinking that the white hare was going for to die
But she slipped out of the bush boys over Howden Hill
And the beagles and the greyhounds they was one short in the kill

There is already some bowdlerisation, though it is at least just a modification of a 'real' verse. The Aspey bowdlerisation is more blatant: a new verse made from scratch and in a different style.

Yes, I still believe that you either sing a song honestly or you leave it alone altogether. No quarrel with people bulking out a fragmentary version with material from other (traditional) examples or from broadsides; but deliberately changing the meaning of a song in order to make singer and/or audience feel comfy and cosy about what might otherwise be a challenging issue is both dishonest and misleading, however well meant.

That isn't what I understand by the much-misused term 'folk process'; this is self-conscious editorial intervention of the kind that early 20th century collectors and publishers are always being criticized for. The 'political correctness' of the Edwardian era was not very different from that in vogue today, though then the problem was sex and nowadays it is other things. They at least had the real excuse that, without their alterations, it would have been impossible to publish material that had even the ghost of a sexual reference. Today, we have fewer such constraints. That doesn't stop people who think they know better from misrepresenting the past by re-writing it, but personally I prefer the truth, even when some of it is disagreeable to modern sensibilities.