Eckard AND ALL
As the DFM solution requires so many variables and cases I itemise each 'case' of light passing between inertial frames, which must be considered as 3D spaces, or different media, 1 and 2, with scattering (refractive boundaries) at each transformation. This may perhaps be a block of ice or gas.(Put the Cartesian system out of your mind) Consider a visible light 'pulse', or short string of photons or waves whichever you prefer.
In all cases, as we know, the light 'signal' the observer sees is scattered from the particles charged by the 'pulse'. (For simplicity we will not consider angles/Cos/theta).
Case 1a) Where the media are NOT co-moving. The observer in medium 1 is at rest with respect to (wrt) both. He sees the light speed and wavelength changed (Doppler shifted) by c/n (refraction co-efficient). But the signal reaching him (eyes or instrument lens) does so via medium 1, so at the 'c' (or local c/n) with respect to the observer. He records c = f*lambda for both the pulse in the other medium and the signal reaching him.
Case 1b) Observer in medium 2. (This may be a gas, i.e. air). All as above. Whether the pulse 'hits' his lense or is scattered to it, he records c = f*lamda, as the light is both slowed and Doppler shifted.
Case 2a); Observer in medium 1. Medium 2 is moving at v wrt medium 1 (say towards the light). Now the observer will see the wavelength the same as medium 1 but the apparent speed (wrt his frame) further reduced, and now ALSO frequency reduced, both proportionally to v. So APPARENT light speed (as it's now in a different inertial frame) = c/n - v. So the laws don't work!? Hmm. We'll go on;
Case 2b); Same, but with medium 2 moving on the SAME vector as the light. We'd get c/n PLUS v. But of course if 'n' is close to 1, and v is larger, that would be superlumenal. But we now see if we NEED the LT to prevent light going faster than 'c' anywhere. Actually of course it it is only 'APPARENT' speed we're seeing, because it's happening in a different frame to the observer. And yet again, none of the laws work as it's moving in a DIFFERENT inertial frame. But that is because it's an INVALID frame for measurement. All we're seeing is an apparent 'rate of change of position' as the light we see is a sequence of scatterings from different particles, emitted at c (or c/n).
Case 3a) Observer IN medium 2. Imagine you're on some planet, out in space, surrounded by an ionosphere, near a star. We'll call it 'Earth', and it's moving towards star b. Entering medium 2 the light from star b is blue shifted TWICE; firstly to c/n in the atmosphere, (as case 1 above), SECONDLY the additional Doppler shift due to the MOTION of the planet. HOWEVER.. In this case, as the SPEED does not change (within medium 2) it is the FREQUENCY that increases, so f balances lamda, to maintain c, So E = f*lamda is invariant, which is the Law of Conservation of energy. c/n = f*lamda is also invariant as this IS the valid observer frame for measurement.
Case 3b) Same as 3a but the planet moving AWAY from the star. Speed c/n stays constant, wavelength increases due to motion and frequency reduces to balance it.
(We can also play with cos*theta etc. for different angles in all these cases but it's only a red herring to confuse clarity of thought here, it does however have a direct analogy in geometrical optics with the time averaged Poynting vector - and the breakdown of the Law of Refraction between co-moving media - but forget all distractions for now!).
Case 4a-z). Observer in motion. This is our 3rd observer condition. He has been at rest in Medium 1 and 2. now he is moving wrt both. In this case there is NO DIFFERENCE in principle to conditions 2 and 3! Light scattered by the gas (air or whatever) re-emitting photons or waves at the local c/n, travels to all observers at 'c/n', (including 'c' in space). If the observer decides to move, he creates a new inertial frame, with his plasma fine structure, and the signal changes to HIS local 'c' when it reaches him, or the glass of his spacecraft or instrument lens. This is precisely what we find! At last! It travels across the space BETWEEN emitter and receiver at 'c', as we know from all the evidence, (as I referred), but it also received at 'c' in the observers frame, as we also know from all the evidence!
Ooops, England just kicking off against Ireland for the Grand Slam!, I haven't checked the above, so do ask...
Peter