Both. Prior to Special Relativity, virtually all scientists believed that light traveled in a medium they called the luniniferous ether. Einstein even referred to this in his 1905 paper. These scientists believed that light traveled in all directions at the speed of c only in this medium. If an observer were moving through this medium and attempted to measure the speed of light, it would be different in different directions. In other words, it would not always be c.
However, when Michelson and Morley did their famous experiment to detect this difference in the speed of light in different directions (since they believed the surface of the Earth could not always be stationary in this medium), they will unable to detect any difference no matter how they rotated their apparatus.
This lead some other brilliant scientists to figure out that if their apparatus was being deformed by its motion through the medium in a particular way, this would account for their inability to detect any change due to different directions. They also figured out that clocks and timers would be affected by their motion through the ether.
Now the way to measure the speed of light is to set up a light source that can emit a very short flash of light, aim the light at a mirror and have the light reflect back near the source where you also have a light detector hooked up to a timer that can measure how long it takes for the light to make a round trip from the source, to the mirror and back to the detector. It turns out, these scientists believed, that no matter how fast you are moving through the ether and how your experiment is oriented, if the shortening of the distance between your source/detector and the mirror is just right and the slowing down of your timer is just right, they will compensate for your measurement of the speed of light in just the right way so that you always get the same answer.
So this is the part that depends on our measurement of the speed of light. If we could determine how to be stationary in the ether, we could make a valid measurement that would not be "rigged" by our apparatus being shortened and our timer running slow.
Note that this kind of apparatus is only capable of measuring the round-trip speed of light, which by definition, always involves light that has traveled in two opposite directions. If we were stationary in the ether, the two halves of the trip would take the exact same length of time but if we believe we are moving in the ether, the two halves of the trip could take different lengths of time depending on our orientation to our motion through the ether. So let's say our apparatus sends the light against the motion of the ether for the first part of the trip and with the motion of the ether for the second part of the trip. The first part will take longer and the second part will take a shorter time, but we have no way of knowing by how much. We can only know the total time.
Now along comes Einstein and he declares that we can just assume that the two parts of the trip take the same length of time (even if they don't really). This is the philosophical part of the speed of light being c. Einstein's second postulate is referring to the one-way speed of light. He arbitrarily defines the speed of light to be the same constant in all directions. In other words, he assumes that any non-accelerating observer is stationary in the ether, even if he isn't. He makes it very clear that we cannot measure this, but from this assumption, and Einstein's first postulate which is that all the laws of physics will be the same for any constantly moving observer, he builds his entire Theory of Special Relativity.
The significant contribution of Einstein's postulates is that it frees us up from being concerned with the state of the ether, we can totally ignore it. You can read about this in Einstein's famous paper:
http://www.fourmilab.ch/etexts/einstein/specrel/www/
)
