Pentcho
I agree, moving observers only change the wavelengths they interact with, they do not affect that in the surrounding frame.
What needs to be done, which is the new part, is to actually apply the finding, which you agreed, that there are two separate states; One before detection, one after. You use the common current assumption it is the one 'before' interaction that matters, I'm suggesting it is the one 'after'. My logic is this;
A detector is a collection of massive particles. If there are no massive particles there is no detector, so can be no detection. We need to think carefully about that first axiom and accept it, or please suggest another logic if you disagree.
If the detector is moving towards the source, then in the time t between wave peaks A and B reaching the detector, the detector will have moved, therefore the spatial point of interaction is different. This is simple kinetics OK?
Then the time between wave 'peaks' in the surrounding medium interacting with a stationary point in that medium is different to the time t' experienced by the detecting particles. Still all OK?
So (assuming index n is the same for simplicity) the distance between the peaks in the detector medium is shorter than the distance in the surrounding medium. OK? That distance is wavelength, lambda. Implicit at that same instant the frequency has changed inversely to lambda. As we know it does in all cases where we are measuring media to media Doppler shifts, including those due to delta n.
So, using relative v and the only 'observable' f in the two media, the detection can tell us TWO things subject to assumptions about the other two values; propagation speed and lambda. It can tell us both the original 'relative' speed, and the new propagation speed, or if we assume a speed it can tell us both the original and new lambda.
Remember, if there is no new medium then there is no detection.
One other entirely different case is there to confuse the innocent. Turn the lens perpendicular to the light. Light passing by is not observable. But something CAN be. If there are gas particles which the light charges on its way past, and the light is quantized, then the particles will 'flash' on and off. The observer can then calculate a speed. Now think carefully. This will be a REAL speed if he is at rest in the surrounding medium, but only and 'apparent' or relative speed if he is in motion.
The massive (pun!) mistake science has been making is confusing that 'apparent' speed with 'real' propagation speed (the light from the 'flashes' actually interacting with the lens is always only doing c).
I agree this seems very difficult to comprehend at first. Once thought through it is entirely self apparent, but there is then the matter (pun2!) of the deeply ingrained assumptions (equivalent to 1,000yrs of flat earth) to overcome. That seems perhaps the far bigger problem.
Do you agree?
Peter