There are three relevant quantities involved here: the length of a meter, the duration of one second, and the speed of light. You only need to absolutely measure one of them, after which the other two can be defined in terms of the one that is measured.
For technological reasons, we have chosen to make the measured reference quantity the length of one second, which is defined in terms of the number of oscillations of radiation associated with the transition between the hyperfine ground states in cesium (specifically, it's 9,192,631,770 oscillations of that light). This is basically because there are experimental techniques that allow incredibly precise measurements of the frequency of radiation, at a level that really can't be matched by length or speed measurements. (The best frequency measurements in the world use trapped aluminum ions as the "clock," and are good to something like one part in $10^{18}$.)
Having defined the second in terms of some physically measurable quantity, we are then free to define the speed of light as having some particular value in meters/second, and then define the meter in terms of the distance traveled by light in one second. The size of a meter is merely a matter of convention, not anything fixed in the physical world, so as long as we have anchored the second to something fundamental, we can make the meter be whatever we want.
The particular values of the meter and the speed of light that we choose are based on older measurements using a meter defined in terms of the circumference of the Earth. We've chosen to keep that value, because it would be a hassle to make a wholesale change.
Probably, you misunderstood the non-absolute Time interval concept. At near $c$, your eyes can't perceive that your time is dilated (and, length is contracted). You and your measurement tools won't feel any difference at near $c$. Your clocks would tick at the same rate for you like that of rest observer.
The only glitch: A rest observer won't be agree with your measured values (of time interval and length) and you won't be agree with theirs. There's nothing to understand here. It's similar to how two different observers don't agree with measured speed.
Best Answer
It's the second one: the reason the speed $299792458\ \mathrm{m/s} = c$ is special is because it's the universal speed limit. Light always travels at the speed $c$, whatever that limit may be.
The reason there is a "universal speed limit" at all has to do with the structure of spacetime. Even in a universe without light, that speed limit would still be there. Or to be more precise: if you took the theoretical description of our universe, and remove light in the most straightforward possible way, it wouldn't affect $c$.
There are many other things that depend on the speed $c$. A particularly important one is that it's the "speed of causality": one event happening at a particular time and place can't affect another event unless there's a way to get from the first event to the second without exceeding that speed. (This is sort of another way of saying it has to do with the structure of spacetime.)