Time dilation has nothing to do with local time or remote time. I'm afraid you misunderstood my comment on another thread:
I was trying to get you to read Einstein's method of synchronizing the time on a remote clock to the time on a local clock as part of the definition of a Frame of Reference where all the clocks are synchronized to the same time and are the basis for Coordinate Time. Since all these clocks remain stationary with respect to one another and to the definition of the spatial coordinates in the Frame of Reference, none of them are time dilated.
If you go on to read the rest of his paper, he eventually gets to the dilation of time on a moving clock, that is, a clock moving with respect to the coordinate system, that is, a clock moving in the Frame of Reference. The time on the moving clock is called Proper Time. The ratio of the Proper Time on a clock moving with respect to the clocks displaying Coordinate Time is the reciprocal of gamma. It's as simple as that.
So please don't say the time dilation is just local time or remote time. You should say that time dilation is what happens to a moving clock (Proper Time) compared to a stationary clock (Coordinate Time).