In reply to stp:
What I am doing wading into another Peak/Peaks debate I don't know but here goes.
I believe that the original name for the region is the Peak (or previously Peake) and 'District' is a more recent addition. 'The Peak' is therefore not an abbreviation or slang.
For example, I came a reference to the 'Peake' in Samual Pepys diary from 19th January 1663: '... and that was all that passed but my Lord did presently pack his lady into the country in Derbyshire, near the
Peake; which is become a proverb at Court, to send a man’s wife to the Devil’s arse a’ Peake, when she vexes him.' (search for 'Peake' in
http://www.pepysdiary.com/diary/1663/01/ ). So perhaps it should be Peake, but it shouldn't be Peaks. Pepys didn't recommend which bouldering guide to use though.
So if anyone can find a reference to 'The Peaks' (or 'Peakes') that predates 1663, I shall humbly start calling it 'the Peaks' (possibly through gritted teeth). But until then, Peak is what I prefer to call it.
Can I rest my case, m'lud?