The myth here is the very belief that these words have any real meaning at all. There are no species or tribes with these names into which people can be divided. I, for one, travel on cycles, on foot (more than on cycles in fact), and in cars (both as a driver and as a passenger). I also travel by train sometimes and even occasionally by tram or bus.
It suits the purposes of some parts of the media – some newspapers, radio shows and television shows – to stoke up rage and controversy by fooling their audiences into believing that these tribes actually exist. Of course, that’s not surprising when their income depends on attracting people to their pages to see adverts.
Cycling, Motoring and Walking are activities that practically anyone could be involved in at any time (and Government ministers ought to know better than to use -ist terms), and the discussion is about how roads and streets are designed so that we, collectively, can choose whatever mode of travel we want to whenever we want to. Even further than that, it is often about how we want to see our streets: as places to spend time, shop, and eat, or as routes to move people and goods, or as places to dump our cars when they are not actually being used.