Hi Russ, I am a bit of a self-taught layman, but this is my take on subtone; you seem to know how it should sound, and the way to get that sound is to take less mouthpiece in your mouth, and play with a more relaxed embouchure and softer tonguing.
As I understand it, a note comprises of the base harmonic, or pure tone, with a combination of various harmonics to give the note it's timbre laid over the ba e tone. Hence when you are playing and changing your tone you are enhancing/emphasising or reducing various harmonics of the basic note (if I remember right this basic note is called the fundanental). And you thought changing your tone was just wiggling your lips, tongue and throat 😉
I suppose an analogy for subtone would be it's like turning down the treble control on your stereo which gives the impression of increasing the lower frequencies.
Also, it's most frequently used on tenor, but also sounds nice on alto, and tends to work best (for me anyway) on the lowest one-and-a-half octaves.
Cheers, Bob