[bcook]

Regarding countersteering & bicycles (and motorcycles), I better understood the phenomenon by picturing a bar shoved horizontally through a basketball.

You can steer by either turning the basketball (horizontal plane) or leaning the basketball (vertical plane). In the case of a motorcycle at speed, the bike's effective turning radius is affected more by the lean than by how much the handle-bars are turned. The handle-bars are more about controlling the amount lean.