For a country with relatively low hills, Britain has an astonishingly colourful history of sorting hills into groups and climbing them accordingly. I think it's due to the bumpiness of our land, the accessibility of nearly every peak, and the trainspotterish tendencies of our people.
It started in 1891 with Sir Hugh Munro's list of 3000ft Scottish mountains. There are now 284 Munros and over 3000 people who have done them all. The Munros are really a subgroup of the Munro Tops, which is the full list. The Munro Tops are all the 3000ft peaks (meaning any peak, not necessarily a distinct mountain) of Scotland, as set out by the SMC, and the Munros are roughly the 284 most important and bold of these.
John Rooke Corbett emerged later with his list of Scottish mountains between 2500ft and 3000ft also having 500ft (152m) of drop on all sides. The SMC took custody of the Corbetts list when he died in 1949. Corbett completed the Munros and Tops in 1930 and is thought to have climbed all the 2000ft mountains of Scotland. He was also in the first party to do all the Welsh 3000s in one go. The 219 Corbetts are much less well known than the Munros. Doing a full round is harder due to the lack of handy ridges with three, four or more to tick off.
In 1992 Fiona Graham published a list of Highland mountains between 2000ft and 2500ft with 150m drop. The list is now run by Alan Dawson and has been extended to include the Scottish Lowlands, making a total of 224 Grahams. Notable Grahams are Suilven in Assynt, Marsco on Skye, and the Pap of Glencoe.
TACit Press runs a list called the Hewitts (Hills in England, Wales or Ireland over Two Thousand feet). Hewitts are 2000ft (610m) high and have 30m of drop all around. There are 525 Hewitts, of which 177 are in England, 136 are in Wales, 211 are in Ireland and N Ireland, and one is in both England and Wales (?!)
Of course it's up to everyone to decide at what point completing lists goes too far; I'd say it's about here, ahead of the following:
The late Arthur Wainwright is famous for his beautifully detailed guidebooks to 214 hills in the Lakes, intended to describe the fells rather than to create a list to be ticked. He chose his hills for their attractiveness and local prominence, not just for their height. Climbing all of these hills would certainly give a good appreciation of the breadth of Lake District scenery.
More recently, John and Anne Nuttall published a comprehensive two-part guide to the English and Welsh mountains using the rule 'over 2000ft with 15m of drop on all sides'. This 15m rule includes many minor outliers.
In my opinion the biggest contribution was Dawson's list of Marilyns which he introduced in 1992 in his book The Relative Hills of Britain. Marilyns are British hills of any height with 150m of drop on all sides. The idea was to list the hilliest hills irrespective of height. The Marilyn list necessarily includes all Corbetts and all Grahams as subsets. There are about 1550 Marilyns (it changes frequently), 6 of which are on St Kilda off the Outer Hebrides.
The drop criterion is here to stay. It might be brought into tables that so far have been random in this regard. The prime example of such a table must be the Munros, which is (as Dawson notes) just a list of 3000ft mountains published by the SMC. TACit Press has published a list called the Murdos which are the 444 3000ft mountains of Scotland with 30m of drop on all sides. This list brings order to the 3000ft mountains of Scotland, but the idea has yet to be made official, so to speak, by the SMC, which is carrying on with the arbitrariness of the Munros. Munros/Murdos with 150m of drop are called (wait for it) Marilyn Munros. Will we see the day when the SMC sorts out the Munros along these lines? The 150m drop criterion would give a solid list of 204 mountains.
It wouldn't be a surprise if a consensus built up as follows: 610m and over is mountain height; 30m drop is a top; and 150m drop is a mountain. In Scotland the Murdos, Corbetts, Grahams and Marilyns make up these groups, with the exception that the Corbetts stipulate 500ft (152m) of drop instead of the regulation 150m. Outside Scotland the groups are taken care of by the Hewitts and Marilyns.
Through the ages, enthusiastic climbers have put together their own lists. Some good ones are known about, and are discussed in some of the TACit Press pamphlets.
I've used a new drop criterion on the Welsh 3000s list that will hopefully prove worthwhile. It's the mountain's drop as a proportion of its height. The rationale for this is that the height by which a peak needs to rise above the surrounding land in order for it to be a separate mountain varies according to the peak's height. Read all about this on the Relative drop page.
Any list based on numbers will miss out some good hills and include some boring ones. There will never be a numerical definition of a mountain.