The book that you have mentioned has the answer to your original question: "What makes the bike turn the same as it was leaned more without hanging off?
It is explained in Chapter 3: Less lean angle requires more effective steering angle in order to keep the same radius of turn (please, see figure 3.18 of page 3-13): "Increasing lean angle tends to increase the effective steering angle."
It is a simple geometrical problem, there is no need to complicate it with camber thrust, slip angles, etc., because the magnitudes of the forces of cornering and the dynamic lean angle remain the same, either or not you hang-off.
The chassis reduces its lean angle when the rider hangs-off while cornering, which changes the relative geometry among the three planes: the ones containing the rear tire, the steered front tire and the curve (track surface).
You may want to do the following experiment:
Fill up a wide recipient with water (the surface of the water will work like the plane of the curve).
Make a central 10-degree bend in a small rectangular piece of cardboard (one side will work like the plane containing the rear tire and the other side like the plane of the front tire).
Keeping the bent edge and both sides vertical, deep the piece of cardboard into the water.
Looking from above, turn the cardboard just like a bike would lean over to turn and note how the angle formed between both lines that intersect the surface of the water and each side of the cardboard gets bigger as the lean angle increases.
That angle is the effective (or kinetic) steering angle, which would force the bike to turn tighter (reduced radius of turn) if the rider would not compensate for this phenomena by steering a little less.
If that experiment still does not convince you, we could use the following well stablished formula:
Radius of turn = [Wheelbase x Cosine of chassis lean angle] / [Steer angle x Cosine of caster angle]
As wheelbase gets a little bit smaller and caster angle remains constant, when the rider hangs off while cornering, the cosine of the chassis lean angle increases (example: cos 45=0.707 and cos 40=0.766).
That change would increase the radius of turn some, making the bike run wide respect to the desired trajectory.
In order to avoid that from happening, the rider must compensate by increasing the steer angle a little.
Another geometrical way to analize that: Imagine a perfectly vertical line running underground by the center of the circular trajectory of the motorcycle.
Disregarding slip and camber thrust, the extended axis of both wheels must intersect with that vertical line.
As those wheels are leaned more, the point of intersection moves deeper into the ground, which reduces the angle formed between the extended axis of both wheels.
Hence, the steering angle must be reduced some in order for the bike to keep tracing the same circular trajectory.
A leaned motorcycle will always have an effective steering angle that is smaller than the one for a 4-wheel vehicle describing the same curve.
The exercise of Motorcycle Gymkhana is a different solution to a problem that is different: make the tightest quick turn around a cone.
The maximum speed at maximum lean angle will make you slower in this particular case, try that experiment as well.
Since speed must be much smaller than during normal Superbike track cornering, the smallest radius of turn of the rear tire is the key to turn the bike 180 degrees as quickly as possible.
For the same reason explained above, the Gymkhana rider wants the chassis to be as leaned as possible during the slowest section of the tight turn.
At full stop lock of the steering, the radius of turn (and the circular trajectory of both tires) will be smaller as the chassis lean angle increases: there is a greater effective steering angle.
Lock the steering of a bicycle at a pronounced angle and push it while at different sustained lean angles for each completed circle and you will see that the smallest circle corresponds with the biggest lean angle.
For the above formula and description of angles, please see "Steering angle" here:
https://en.wikipedia.org/wiki/Bicycle_and_motorcycle_dynamics