I promise not to get impatient so long as we keep somewhere withing sight of the path.
you wrote:
"I never say "a clock measures time".
But you should. That is precisely what it does.
What a clock does is measure the number of times a specified event occurs within a measured period. It might be the pendulum swinging back and forth. Or it might be the number of times a Cesium atom jumps from one energy level to another.
What makes one clock more accurate than another is if you build a bunch of them and put them together ... the more accurate ones stay most closely synchronized.
So lets return to your original premise which was two clocks ticking at rates due to acceleration.
What we are saying is not that the clocks change. Each clock will record exactly the same number of transformations of a cesium atom per minute, or hour, or whatever.
What will change, relatively between the two clocks, is that the number of transformations will be greater for one than the other. In a one hour period one clock might have x transformations whereas the other, perhaps, 1.1x transformations. But to each person observing their clock nothing will change. Only someone observing both clocks will observe a difference.
Hope this helps.
