Let's put it this way: there is somebody traveling in a ship at a certain speed, and another standing on the ground of a planet, which is also moving at a certain speed. Both of these people can say that they themselves are inertial, and it is the other that is moving, and both points of view would be valid.
Now imagine the ship is zooming by the planet with it's light clock turned on. On board the ship, the beam of light is moving in along a path that is at a perfect 90 degree angle to each mirror. However, to the observer on the ground, that beam of light is moving at a different angle (in order to keep up with everything else aboard the ship), and therefore the light must travel a farther distance between each of the mirrors.
However, even though they cannot agree on how long it takes for the light to bounce off of each mirror, they can both agree that the light is moving at the full speed of light.
Length contraction also comes into play here, which explains why both observers will measure their own light clocks at the full speed of light. The faster one travels, the more contracted their length becomes, as well as the length of everything else within their own frame of reference (including any and all measuring devices). Therefore, the moving observer will not notice this length contraction at all.
So, both observers will argue on how long the light beam took to travel between each mirror, but they will both agree that the light beam was moving at exactly the full speed of light.