As for the "correct" way to do a shake, I'm probably way off in what I do. By the way, I had no idea that there were "East Coast" and "West Coast" versions of doing a shake. I just figured that what you did on the shake depended on what was required musically - sometimes a tight fast "shake" or trill is needed, (trilling between two adjacent partials) and sometimes a wide shake (up to a 5th) makes for a really nice effect.
Anyway, what I have been doing for years, I discovered is pretty close to how Don Jacoby describes it in his book. I set my tongue forward and arch it, which pushes the note to the point where it's just about to "break" to the next higher partial. Then, by slightly increasing the mouthpiece pressure just a bit, it causes the note to "break" and move up to the next partial. So basically, I use my fingertips on the valves and "shake" the horn forward and backward, which causes a trilling or "shake" effect. Of course, all of this happens pretty fast and I don't really think about how I do it too much anymore, I just sort of do it.
New Monopoly plays an arrangement of "Devil In A Blue Dress/Good Golly Miss Molly" and there is an 'A' in a line that I like to shake, and the line is played three or four times. On all but the last time through, I just shake it fast between two partials. The last time through, I shake it as wide as I can - either a 4th or 5th - it's not written, but it just seems to fit.
For me, the shake is one of those things that I use pretty sparingly because for the most part, on much of the music we play, it just doesn't fit.
Back to the subject of how to do it, does Maynard actually shake the horn, or is he doing it all with his chops? I know guys who can do outrageous shakes and they don't shake the horn at all. The reason I shake is because I usually can't get them fast enough just using my chops.