Thanks for the answer, interesting.
Sorry for that, but it was a quick answer for me, I had to go sleeping.
Now again, so I wanted always answer later.
But I can already give a quick answer (I come back later on it in the weekend).
Your situation is mine with the car but with light in an angel right on the car's top. If I don't believe in time dilation there is nothing special for me in your example. In both cases the clock has the same rate but specifies a different time because its light path is shorter.
But you can also ask, why can I explain in the moving direction with length contraction and so a shorter meter (1/γ) the same time 1/γ.t with and without time dilation (Lorentz) ? Just as you always say "you need a clock to measure time", so I say "you need a ruler to measure a distance" ...
I go also to look to your other example later ..