I think the original definition used in the UK was intended to distinguish between incidents that arose because of controller issued an incorrect clearance to enter the runway (i.e. ATC error) and those where the pilot/driver incorrectly believed they were cleared to use the runway. The reasoning being that the two types of error occur for different reasons and need to be addressed in different ways.
The earlier definition was also developed before there was any consistent description of what a runway incursion was - there were many competing definitions, all broadly the same. Later ICAO developed an international 'standard' definition, which was adopted by Eurocontrol for use in its work across the region, and it seems that all the CAA is now doing is harmonising with the more widely adopted definition.
What is more interesting is that the phrase 'the protected area of a surface designated for the landing and take-off of aircraft' remains in the ICAO definition. But I've never been sure what the protected area is.
In reality, as already suggested, it doesn't make any difference in practise because it's intended to allow data on the numbers (and, I guess, ultimately, the causes) of runway incursions to be monitored and compared against a consistent baseline........and if you die because of a collision on a runway the statistics aren't going to matter to you.