Without doubt, a certain amount of motor vehicle use, away from the tarmac, is due to ignorance and not deliberate law breaking. Some, however, do flout the law in the hope that there will be no enforcement.
There is no gray area. Unless you are using the motor vehicle on private land, with the permission of the landowner, both vehicle and driver/rider need to comply with the law.
It is illegal to use a motor vehicle on a footpath, bridleway or restricted byway. A byway open to all traffic (take a look at the Rights of Way Information on an Ordnance Survey map) is a public carriageway, requiring exactly the same as the main roads through a town or city.
You will need a driving licence and insurance (for the vehicle you are using). The assumption being that the driver/rider is old enough to have both.
The vehicle will need to be registered (number plates) with a current road fund (tax disc). It will need road legal tyres, road legal exhaust and and MoT certificate, where applicable.
Mini-motos and competition bikes can only legally be used on private land. The beech or the park are out of bounds. The penalties can be harsh, so think twice before letting your child loose in a public place on a motorised vehicle.
The same applies to motorised scooters and powered skateboards.
Be warned, more rigorous enforcement has been promised by Government.