"The speed of light is constant, no matter how an observer moves relative to the light."
This is true. The light is moving independently, thus an observers actions of movement, make no changes to it.
However, the interesting part is that the speed of light is always "Measured" as the speed of light no matter how an observer is moving relative to that light. This includes an observers movement in any particular direction, meaning any direction relative to the direction of the light itself.
The key to understanding this is to view reality in a 4 dimensional manner.
All mass particles are constantly in motion at the speed of light within that 4 dimensional environment known as Space-Time. All that can be done is change the direction of that constant motion. Thus if one is at true rest in space, the entire constant motion is now across the dimension of time only.
To perform a measurement, one must have a complete measurement instrument. This will include one instrument that measures spatial length (Ex. Ruler ). Two synchronized clocks are also required. The clocks are placed at the opposite ends of the ruler.
If light, from any source, passed from one end of the ruler to the other in either direction, while the ruler is at true rest in space, a measurement of the speed of that light will indicate that the light traveled at the speed of 300,000 kps.
If the measurement instrument was now in motion across space, then a change of direction of constant motion across Space-Time has occurred.
In turn, there is a reduction of motion across the dimension of time, thus both clocks are now ticking slower. Rotation has also taken place, thus the ruler begins to extend across the dimension of Time and thus less of it extends across Space, hence we encounter spatial ruler length contraction.
Also, due to this extension across time, the clocks at the ends of the ruler are no longer synchronized, since they are now positioned at different points in time.
If once again, light from any source passed from one end of the ruler to the other in either direction, a measurement of the speed of that light will still indicate that the light traveled at the speed of 300,000 kps. This is due to the multiple changes of the measurement instrument itself.
Thus in both cases we measure the speed of light, as still being the speed of light.
Thus in turn we can not determine if we are at true rest in space or not, since the results are the same whether we are at true rest, or not. Thus we also can not determine what direction we are traveling in within Space-Time.
On top of that, if we had 2 identical spaceships, with one being at true rest in space and the other being in motion across space, to those onboard the one at rest, it seems as though the other spaceship has shrunk in spatial length, and that their onboard clocks are ticking slower.
However, due to the change of the measurement instruments onboard the moving spaceship, it appears to them as though the space ship that is at rest in space has shrunk in spatial length, and that its clocks are ticking slower. Thus from their point of view, it appears to be the spaceship at rest that is actually in motion, rather than their's.
What is important is to understand why in both cases it always seems to be the other guy who has shrunk in length, and the other guy whose clocks are ticking slower.
This rule applies to two bodies, be it one at rest in space and one not, or two bodies that are both moving across space but moving a different velocities.
Thus the absolute point of view can not be detected, thus absolute was dropped from being seen as important, thus, in most cases these days, violent opposition continues against the absolute even though such an intuitive 4 dimensional point of view explains it all and produces the exact same equations.