The problem is that we are considering where it is legal to ride baased on what a track is called, e.g. (quoting from the page linked above)
Types of public rights of way
England has about 190,000 km (118,000 miles) of public rights of way, providing many opportunities to enjoy the natural environment.
footpath-yellow
Footpath – open to walkers only, waymarked with a yellow arrow
bridleway-blue
Bridleway – open to walkers, horseriders and cyclists waymarked with a blue arrow
restricted-purple
Restricted byway – open to walkers, cyclists, horse riders and horse drawn vehicles. Waymarked with a plum coloured arrow.
In fact a footpath is a trail where the right to pass on foot is recorded on the definitive map, likewise a bridleway is a trail where the rights of horses and walkers(and cyclists by dint of a subsequent law) are recorded. The fact that a trail is recorded as a footpath DOES NOT MEAN that a cyclist has no rights: merely that they have not been recorded. As Samuri says, if a law has been passed stating that cyclists are forbidden from riding on the trail, a sign will be displayed.
note that footpath in this sense does not mean the pavement at the side of the road