There is also Shakespeare Cliff. I went up there on the longest day and had the whole place to myself and the mutt.
Park the car in Aycliff, or take the bus, near to the footbridge over the A20, over said footbridge, throughg the gate and there you are. The place is fenced top and bottom, so safe from cars, and 99% safe from the cliff edge. Plenty of space for the dog to run about in too, plus fantastic views.