Speaking for myself (and disregarding any 'official' definition), I tend to agree with Wainwright. In his introductory remarks on Scafell Pike, AW writes:
"The difference between a hill and a mountain depends on appearance, not on altitude (whatever learned authorities may say to the contrary) and is thus arbitary and a matter of personal opinion. Grass predominates on a hill, rock on a mountain. A hill is smooth, a mountain rough. <snip> Roughness and ruggedness are the necessary attributes ..."
Recently, I visited Skye and on one of the days went up the Great Stone Chute to Sgurr Alasdair. I know the range is called the Cuillin Hills but if "roughness and ruggedness are the necessary attributes" then as far as I'm concerned the Cuillins are indeed mountains.
Similarly, standing in the amphitheatre of Hollowstones gazing in awe at the grandeur of the towering crags below Scafell, 'mountain' is the word that springs to my mind. Rock scenery like that is simply not my idea of a 'hill'.
The Cotswolds, the Chilterns, the Mendips - those are hills.