When a cyclist rides with no hands and shifts the hips to turn, what actually turns the front wheel?

Answer: As the bike leans, the steering geometry and the front wheel's gyroscopic effect both turn it into the lean

The rider's body lean twists the fork and front wheel directly — Body lean cannot twist the fork by itself. The rider is connected to the fork only through the frame and a steering bearing that turns freely, so shifting the hips tips the whole bike without wrenching the front wheel sideways. What the body does is start the lean; the fork then turns on its own, driven by the geometry and the spinning wheel.

As the bike leans, the steering geometry and the front wheel's gyroscopic effect both turn it into the lean ✓ — Correct! A bicycle's front wheel is mounted with 'trail': the steering axis meets the ground ahead of the tyre's contact patch, so once the bike leans, the weight on the front wheel turns it into the lean by itself. The spinning wheel adds a gyroscopic nudge in the same direction: tilting a spinning wheel makes it turn about a perpendicular axis, into the lean. Between them the wheel points into the turn, the bike follows a curve, and the rider's hands are not needed at all.

The rear wheel pushes the frame sideways and drags the front wheel around — The rear wheel is fixed in line with the frame and cannot steer, so it cannot drag the front wheel around. It simply follows the front wheel's track; the sideways push you feel in a turn comes from the tyres gripping the road after the front wheel has already turned.