Glute shimmy: squeeze the right, squeeze the left, right, left, right, left. Make sure to relax each one while the other "fires."

I could be wrong, and others will correct me, but the glute shimmy is often a touch "slower" than a piston shimmy

I'm not familiar with "resting" shimmy. I'm not sure if that's the piston shimmy or not.
What I was taught as a vibration shimmy is severely fatiguing. Basically you tighten every muscle in your body all at once, as hard as you can until everything has a small vibration.

But the piston shimmy can be small and look "vibratey."

You might want to check with the Real Suhaila tribe about this, because there are definitely better and more advanced students in her format than I would can give you more and perhaps more accurate information re: glutes.

That said, in Suhaila's format, there isn't a "shimmy" per se. Rather, there are alternating glute contractions that are timed with the meter of the music. They begin to look like a shimmy at double time (if you perform double time alternating glute squeezes with the downbeat on the right glute, your glutes would be contracting RLRLRLRLRLRLRLRL on 1 e & a 2 e & a 3 e & a 4 e & a--that is, the right glute would contract twice per beat, on the 1 and the &, and the left glute would be alternating with the right, and so contracting on the e and the a of each beat).

The Suhaila vibration, meanwhile, is incredibly difficult to accomplish. Long story short, it involves isolating and contracting your gluteus muscles hard enough that they begin to vibrate. It takes a LONG (read: usually years) to achieve, at least so that it is big enough to be seen by an audience.

There are many different methods of vibration, however. I've encountered teachers and workshop instructors who teach/perform them as a held contraction of all the muscles in the leg (including quads, hamstrings, and glutes), or just with an isolated, held contraction of the inner thighs. It depends on what look or stylization you want to achieve--and of course, check with somebody who knows about anatomy and physiology before you start practicing them all the time. I have no idea if other (non-glute) vibrations are good or bad or neutral for your body.

As for the original poster, tightening and toning your glutes shouldn't mess up your shimmies. However, there are methods of shimmying in which you want to keep your glutes loose. When I started studying with an "Egyptian style" teacher, it was really hard for me to learn to let my glutes go, I was so used to using them to power my movements. Because of that, my shimmies looked stunted, but it wasn't because of the conditioning. One thing that you DO want to watch out for, though, is your IT bands. I found out a couple of weeks ago in a Suhaila workshop that strengthening the glutes can cause your IT bands to tighten up, so you need to be diligent about rolling them out (using a foam roller). A lot of activities cause your IT bands to get tight (running, ballet, etc.)--rolling is ALWAYS a good idea.