Conservation of angular momentum GIF

[mklotz]

"An externally-applied torque (the standing man spinning the wheel by hand) produces vertical angular momentum in the wheel."

The angular momentum is the scalar moment of inertia of the wheel times the vector angular velocity of the wheel and is thus a vector aligned with the vector direction of the wheel's spin along the axle axis. So when he spins it the angular momentum is horizontal, not vertical.

The seated man is using the bicycle wheel as a reaction wheel to torque himself around exactly as the telescopes (and other satellites) use their reaction wheels to orient themselves.

Reaction wheels produce a smoother motion than control jets and have the advantage that they never run out of fuel as long as the solar paddles can keep charging the batteries.