To get back to the original question ---
The whole time-dilation business came out of Einstein's solution to a problem that had nothing to do with clocks. Physicists had a problem in as much as they had been unable to find any fixed frame of reference against which to measure absolute velocity. There was no possible way to find out who was moving and who was stationary.
Einstein's fundamental postulate was that the laws of physics were the same for all inertial observers, an inertial observer being one who is travelling at constant velocity. If you couldn't tell who was moving, you couldn't tell who was getting the right answers so maybe they all are. It was a very plausible assumption but it had some unexpected consequences.
Question: What happens when two observers try to measure the speed of the same light beam? The light is travelling from the back of a moving train to the front. One observer is on the train while the other is on the ground. Common sense dictates that they must get different answers. The train will have moved forwards while the light makes its trip and so, as seen from the ground, the light must travel further.
Problem: Maxwell had already shown that the speed of light could be derived in a simple way from two physical constants, namely the permeability and permittivity of free space. Both of these constants can be measured in the lab and so, according to Einstein, both observers must get the same answers.
To get round this paradox, Einstein had to throw away the idea of absolute time.
This was the only way to make the speed of light the same for all inertial observers. As seen from the ground, the clock on the train appears to run slow, and not just because it's moving. If you set up a pair of synchronized clocks along the track, you can check the moving clock against each one as it passes. It will be running slow. On the other hand, as seen from the train, a clock on the ground will run slow.
It gets even more complicated because, ridiculous though it may seem, a moving object shrinks in the direction of motion. If you measure the length of the train as it passes, it will be shorter than if it was stopped. Moreover, as seen from the train, a station will also appear shorter as it goes past.
When all of this is put together, something else drops out because the old law of conservation of momentum doesn't quite work. We can either abandon the law - which no physicist wanted to do - or modify the definition of momentum. Instead of P = m x v, it becomes
P = m x v /sqrt(1 - (v/c)^2)
If you look at this equation, you will see that momentum tends to infinity as you approach light speed. This effect is usually described by the rather dubious claim that "the mass goes up". If this forum was better suited to writing mathematical equations - and if I could be bothered - I could go on to derive what is probably the most famous formula in physics: E = mc^2.