No! It isn't the frequency of the hyperfine transitions, it's the frequency of the emitted light. Think of one electron dropping to a lower energy state. That's one event. That one event has no frequency. But emitted light does.
You are correct, at least in part above. I got caught up in your objection to the use of the word frequency. The cesium clock rate is based on the microwave transmission, but there is also a hyperfine electron transition occurring and it does have an associated frequency. If it occurred only one way, as a single transition we would not get enough data for the clock to be of any use.
The point is that when you are defining the second using some light, you cannot say "this light has a frequency of x" because frequency is cycles per second and you haven't defined the second yet.
This is circular reasoning. Others have already elaborated on the fact that a second was defined before cesium clocks were available. The second did not change, its definition was just refined, to a more accurate standard. If this were not true your position would suggest that temperature or magnetic fields, cause time dilation because they affect the microwave emission of the cesium atom.
Have no doubt that gravitational time dilation is proven.
I never said that evidence we now have does not support time dilation. What I said or intended was that time dilation had not been proven in any universal context.
Farsight, what happens to the microwave frequency associated with a hyperfine transition of a cesium atom if you change anything about its environment? Say, temperature or magnetic field... The clock rate of atomic clocks can be affected by more than just location in a gravity well. I accept that the experimental evidence demonstrates that the clock rate of an atomic clock based on hyperfine electron transitions, is time dilated in a manner consistent with the predictions of GR. That does not prove that all change is likewise time dilated in an equally consistent manner. For time dilation to be proven, universally it must be demonstrated to affect more than one type of change in a manner consistent with prediction.
My original objection to your use of the parallel mirror light clocks, was and is that you use it as if it is some evidence of fact. It is a hypothetical clock and remains a hypothetical clock no matter who or how many "people" believe the underlying assumptions and theory, and/or that the hypothetical can be used to communicate the theory...
Using that hypothetical as some sort of proof is like using the bowling ball analogy as a proof of gravity curving spacetime. You cannot with today's technology build a light clock with the accuracy of the real optical clocks available... And your use of the hypothetical parallel mirror light clock is based on the assumption the the speed of light or distances change in a gravity well. That is not how it is used in the references cited by your Wiki reference.
To be honest I don't know if the speed of light is universally constant or not. I understand some of the arguments on both sides of the debate, but so far it has not been accurately measured except in the essentially uniform inertial frame we experience in our earth bound laboratories.