When is a peak a mountain in its own right?
Take Snowdon and its neighbour Carnedd Ugain as examples. Everyone would think of Snowdon as a mountain in its own right, but is Carnedd Ugain? After all, it's only an easy 75m plod up from the lowest point between it and Snowdon. This 75m figure is Carnedd Ugain's drop. Drop is also called relative height in the UK and prominence in the US. The essential point is that we want to disregard those peaks which are just a shoulder on the side of a higher mountain, but celebrate those peaks which are prominent from all sides, rising up proudly and needing a big effort to reach them.
In the Lakes there are over 200 peaks of 600m in height. At one extreme we could say that all these 200+ peaks are mountains; at the other extreme we'd say that only the highest peak, Scafell Pike, was a mountain. As soon as we say 'this peak qualifies too because you have to climb n metres when walking to it from higher ground' then we have to choose n, which is the whole problem.
The commonly used definition is:
A peak's drop is the difference between the height of that peak and the height of the highest col connecting that peak to any higher peak in the region.
Sub-definitions: a peak is a point from which the ground slopes down, or along and then down, in all directions; a region is a contiguous group of two or more peaks. If a peak is the highest in its region then it must take its drop measure from a higher peak in another region. This other region might be next door, or at the other end of the land mass. In the case of island or continent high points, the higher peak must be on another island or continent, necessitating a drop to sea level to climb them from that higher peak, meaning that their drop is equal to their height.
In practice, we divide peaks up into massifs, where we can define a massif as a group of contiguous peaks with a valley (loosley defined - I'm sure a watertight definition is possible) all around. The highest peak in a massif measures its drop to a higher peak in another massif, and all the other peaks in the massif measure their drop to higher ground surrounding them within the massif. Usually it is clear from maps how much drop is involved, but when a peak has two or more higher peaks next to it, it takes its cue from the peak to which it has the highest col, even if this peak is lower than the other peaks.
An alternative definition of drop has gained popularity in the US:
A peak's drop is the difference between the height of that peak and the height of the highest contour line encircling that peak and no other peak.
This is an intuitive and appealing definition, especially when you see an aerial view of mountains in your mind's eye.
Drop often understates the climbing needed
The drop measure will understate the climbing needed to reach a peak, when there are undulations on the way. Drop refers only to the peak's height and the highest linking col's height. Between these two heights there can be any number of dips needing re-ascent.
Distance as an alternative to drop as a criterion
The theory behind using distance from a parent peak as a measure of a peak's independence is that a peak will be a peak in its own right if it's far enough away. An example of a list based on distance is Percy Donald's eponymous list of mountains in the Lowlands of Scotland which used a complicated formula involving both distance and drop to determine the worthy peaks. Distance seems an even more arbitrary measure than height or drop; it's unlikely to figure much in future.