This is all wrong. I never had much interest in magnetics, but I am very interested in precision oscillators.
So as I was drifting off to sleep, I was trying to visualize the fields coming from the windings. I ended up totally confusing myself.
Now, I believe drilling a hole in the side of the toroid offers significant benefits. The hole is easier to drill, and it allows two separate windings. These can be wound in series-opposing, which cancels the effect on the main winding.
For example, with a single winding as in the original, if you measure the inductance with the winding open and shorted, you should get two different inductance readings.
However, with two separate windings in series-opposing, having the windings open or shorted should have no effect on the inductance.
This is extremely important in the case of an oscillator. You need the highest Q possible for the lowest noise. If the control winding loads the inductor it will reduce the Q. But with two separate windings in series- opposing, it should not load the inductor and consequently have no effect on the Q.