Gain twist rate is measured by the twist rate at the muzzle, and has no particular significance in regard to GS.
Gyroscopic stability factor increases with range because the rate of linear velocity decays MUCH more quickly than RPM, and aerodynamic forces are likewise diminishing.
The illustration of the static but spinning top is not pertinent. Spitzer bullets, both flat base and boat tail designs, generally have CG aft of CP(center of aerodynamic pressure), and this is the reason that they have to be stabilized by twist in the rifling. LONGER bullets, such as boat tails and copper alloy designs generally have a greater distance betwen the two, thus requiring faster twists. It is noted that even round balls benefit from a small amount of twist, as are all aerodynamically stable projectiles, but there are only three ways I'm aware of to alter the equation for GS. 1)core density, 2) moving the CG, and 3)changing the diameter of the bullet.
Item 1: using lead as a basis for comparison, tungsten is denser, copper less so. Item 2 might be accomplished by having a low density base core such as aluminum, copper, or plastic. In 3, Angular Momentum is changed in this example because at a GIVEN RATE of rotation, there is more angular velocity/energy in a larger caliber bullet and less in a smaller one. The energy change is NOT linear to diameter. GS is all about angular momentum vs the overturning moment of aerodynamic forces. Items 1 and 3 involve change in angular momentum, while number 2 will affect the distance between CG and CP, thus changing the amount of AM required for stability.
I'm not a rocket scientists but can refer you to a couple if you desire(books). It is perhaps easier to understand if you look at the math formulas and give it a bit of thought. I don't find it easy to explain in discussion.
Maybe that'll help.