Interesting. I’m still very much a novice on whistle, but one thing I’ve been doing automatically with my whistles is to change how I blow and how I shape my mouth cavity, and more. In order to take control of the tone. This is just what I do withouth thinking, because I’ve been playing harmonica for most of my life and of course with harmonica how you blow, and, in particular, mouth cavity shape, is everything..
The effect on a whistle is of course minimal compared to a harmonica, but I’ve just been doing this without thinking. I haven’t consciously checked if it “works”. But I know that I don’t worry much about intonation - if I play with someone else then it will be in tune, at least.
[Edit: To add: consciously I’ve thought that how you blow (except for force) can’t matter with a fipple instrument, but uncounciously I’ve believed differently. If that makes any sense.]
Interestingly enough, I am also a beginner, but thought that adapting one’s embouchure to get the desired effect was a commonplace for all whistlers. Apparently, I was wrong 
For instance, in order to flip the octave, I don’t blow much harder, but instead raise the back of my tongue slightly towards my palate. I tried simply blowing harder, but it takes much more effort to do it that way.
Come to think, I believe Grey Larsen, in one of his meticulously written books, also suggests that embouchure matters. I’ll look that one up. 
I’m not sure that qualifies. In addition to mouth shape, he changed the entrance to the windway:
Yes, thanks, that’s the one. Clips below:
I think one must agree that the differences are substantial.
I’m not sure why you’re not sure. If the question is about what you do with your mouth, what’s to exclude?
If you obstruct part of the windway, whether with your lips, poster putty or the remains of your lunch, it will alter the geometry of the whistle, so I’d expect it to affect the tone. The OP was about, “the size of [Mike’s] mouth cavity.” I’m curious what can be done without direct interference with the air stream.
Fair enough.