My understanding of that is you cannot enter the byway if you are going to stop for camping sleeping or cooking. Enduro bikes and bikes should be ok.
prohibition of entry to all
vehicles (which includes caravan, caravanette or like vehicles, horse box, trailer, carriage or horse
and carriage) for any purpose including sleeping, camping and cooking on Byway 11 Amesbury,
Byway 12 Amesbury (south), Byway 12 Amesbury (north) at their junctions with the A303.
But the any purpose is a bit worrying