I'm a little confused by the following classic example about the speed of light being constant for all observers (paraphrasing):
Jack and Jill are both travelling at a significant portion of the speed of light; Jack is travelling behind Jill. Jack shines a laser pointer at Jill. Both observe the light emitted from it as travelling at the speed of light, $c$.
I understand that the speed of the emitter of the light doesn't influence the speed of light; the photons always travel at the maximum speed they can. If Jack were moving backwards, the photons would "instantly accelerate" to $c$; Jack moving backwards wouldn't detract from their speed. Jack moving forwards isn't adding to their speed, because they can't move faster than $c$.
However, the observer's perspective is harder to understand for me. Assuming Jill is observing the incoming laser by counting the frequency of incoming photons, shouldn't that rate change depending on how fast Jill is moving herself? Assuming Jill would be travelling at the speed of light (ignoring the practical impossibility), shouldn't she not be receiving any light at all? Scaling that down to her travelling at half $c$, shouldn't her incoming photon rate be half as much as if she was "standing still"?
(This is very related to Seeing light travelling at the speed of light, however it's about the other side of the same setup.)