Your last point doesn't explain why or how time dilation occurs, just that we know that the speed of light is a constant to all reference frames.
There is a famous experiment with three synchronized atomic clocks. Two were flown in opposite directions around the globe the other stationary. The eastward flown clock lost 59ns and the westward flown clock gained 273ns, both relative to the stationary clock. The reason is that the clock flown eastward is travelling faster since it is in the direction of the earth's rotation. Measurements are consistent with the theory of relativity.
https://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experim...
But that's what the light clock experiment [1] is all about. The two observers see the light moving at the same speed, but one observer thinks the light takes a longer path than the other observer does. If both observations are equally valid, the inevitable conclusion is time dilation.
[1] http://www.emc2-explained.info/The-Light-Clock/#.WBxO8vnLdPA
It makes absolute sense for my gas tank to be lower if I take a long roundabout path from A to B while you take a short one, even though our cars have no idea about what the other one is doing.
I think in some sense this is the wrong question to ask. Your watch isn't running slower. The point is that if you're moving, you must be moving relative to something else. Their watch will appear slower to you, and your watch will appear slower to them. The counterintuitive part is that you're both correct.
The reason all this comes about is that both observers measure the speed of light as traveling at the same speed, but they can't agree on the path that the light has taken.