The issue with time dilation descriptions like the one with the train and embankment is that they often miss a critical point, since they generally focus on the embankment observer alone.
So let's consider both observers.
We start with two colocated light clocks, 1 stationary to our reference frame and one in motion. In these animations, the yellow dot is the pulse of light bouncing between the mirrors. The expanding circles are radiating outward at c, and act as reference showing that each pulse is moving at c relative to the frame of reference.
Animation 1 is the from the frame in which the red clock is stationary.
As can be seen, the red light clock ticks faster than the blue clock. If, for instance, the round trip for the red clock takes 1 micro second, it takes longer than that for the blue clock to complete one round trip. But what if there were someone "riding along" with the Blue clock? What would be happening according to them? This is what Animation 2 shows
Since light travels at c in all inertial reference frames, In this frame, it is the Blue clock that takes 1 microsecond per round trip, and the red clock that ticks slower.
Keep in mind, we have changed nothing from the previous animation other than switching observers. And there is no reason to prefer Red's perspective of events over Blue's or vice-versa. Both are equally valid. The two frames just measure time differently. This is the essence of Relativity.