So I had my headlights out of the car yesterday to refinish the lenses as well as check out everything I possibly could with respect to how they are mounted to try to reduce the shaking. While I had them out I removed the back panel used for bulb replacement and thoroughly looked and fiddled around inside.
Both horizontal and vertical adjusters are intact and connected to the assemblies. The servo motor (non-adaptive, auto-leveling) is also intact and firmly connected. No internal components appear loose. There is some movement in the internal assembly if you push the bulb socket around but it isn't loose and rattlely, there is just some play because of the adjusters and auto-leveling linkages.
I studied how the light and frame mount to the body and tried some different positions. Everything leads to "sproing" when you thump on the assembly. I think this is 100% my problem and nothing is wrong inside of the lights. None of the mountings for the lights or light frame seem to be missing, damaged or improperly assembled.
Why is my car special? Both lights too, not just one.
I tried to take cell phone video of the problem with the bumper off and everything clearly visible. I haven't viewed it on PC but I believe that the reduced frame rate (15 fps I think) basically makes the vibration unviewable since it's too fast to be captures by 15 fps. I'll check it out and see as soon as I have time.