Peter,
"why is a Doppler shift not observed on Earth for light flashed terrestrially despite Earth moving in a direction towards or away from the flash?". Because Earth, it's atmosphere and it's ionosphere clearly represent an "Inertial system" (mass all in one definable state of motion; orbiting the sun at a velocity v),...
CORRECT, thy mouth sayeth it, but using so many words. Lesson to take away: With an earth-bound medium (whatever pet medium individuals come up with), the Michelson-Morley experiment on earth surface will not reveal the earth's motion by using light. No need for length contraction and time-dilation. No need to continue your romance with SR. (It is important to note however that the pet medium must exist in the terrestrial 'vacuum' since the M-M experiment gives same result).
On the analogy of 'TWO cases of emitter/observer motion' is simply described by analogy with sound.
In case 1) you drive towards me and beep your horn. Both I and (R) hear the sound Doppler shifted to a higher pitch (shorter wavelength, NO and higher observed frequency, YES.
In case 2) I drive towards you (and (R). Now the sound is approaching me at s+v wrt MY rest frame. I still hear your horn Doppler shifted to a higher pitch. but DOES (R)!? YES, R will also still hear my horn at same higher pitch, but you will hear at a higher pitch than the higher one you heard in case 1).
But you ask; "as it left the car it was doing c wrt the car, so where did it change speed!?"
I am not sure I get your meaning here. The sound waves DID NOT EXIST somewhere before hitting the medium. The waves are a disturbance in the medium. Very simple, so "no 'extinction distance' in air, which started immediately the sound waves hit the medium".
On the frequent use/ abuse of absorption and re-emission,...
Perhaps you don't take cognizance that if things happen that way, when an electron or whatever particle absorbs the light, the re-emission will be in ALL directions and not necessarily in the direction of initial propagation. If we are to believe this, then a light ray sent on a message to deliver a certain amount of energy to a receptor will have almost all its energy re-emitted and sent to different unintended destinations. I don't think that is what is observed. Lasers tell us this and you can take this as a guided missile against this part of DFM! What is observed is that some of the encounters on the way may retain only some of the energy, if it is matter. Most of it gets delivered to destination if it is in empty space.
Finally, the correct and universal Doppler equation would have to be
f' = f(1 + u/c), where u is the relative velocity of observer and source to each other, with + or - sign being applied appropriately. Without this an observer will be able to tell who is doing the moving between source or receptor because f(1 + u/c) is not equal to f [1/(1- u/c)]. There is a difference of u22/c2 as I pointed out.
Best wishes,
Akinbo
*Laptop battery running low...