Iām having difficulty gauging your understanding, based on what is written here.
Are you thinking that (kinematic) time dilation is merely an optical effect, and produces no measurable physical consequences other than what an observer can visually see?
He perceives it as dilated (āslowedā), because they are in relative motion with respect to one another.
He also perceives it as dilated, because they are likewise in relative motion. In the formula for kinematic time dilation, the relative speed appears squared, so its relative sign (moving towards or away) is irrelevant.
He perceives both A and B to be dilated (slower), because he finds himself in relative motion with respect to both those frames. He perceives his own watch at C to be ticking normally (no dilation), because there is no relative motion between himself and his watch. They are in the same frame.
No, because heās in the same frame as that clock, so there is no relative motion. Kinematic time dilation arises due to relative motion between frames.
Observer C himself notices nothing special - his own clock ticks at 1 second per second from his own point of view (no relative motion). However, he sees both A and B going slower - and conversely both A and B see C ticking slower from their own vantage points. Thatās because in the frame of the train, both A and B are in motion whereas the train appears stationary; whereas in frames A and B, the train in frame C is in motion, whereas A/B are stationary. In both cases, the respective observer sees the other clock to be in motion, and thus dilated. The observers just trade places.
Because kinematic time dilation isnāt something absolute that āhappensā locally to a clock - it is a relationship between frames/clocks.
Think about it - from the vantage point A, the train is in relative motion with some constant speed v, whereas A itself appears stationary. From vantage point of the train on the other hand, frame A is in relative motion with that same speed, whereas the train appears stationary. In both cases the relationship between the frames is the same one - relative motion at speed v - so they both see the same thing, namely the other frameās clock being dilated. This is also exactly what the mathematics tell you. The relationship between frames is the same one irrespective of which frame you find yourself in - thereās the same relative motion (v is always the same), thus in each case the clock thatās seen to be moving is dilated with respect to the observer, and never appears to be speeding up; youāre plugging the same v into the same formula to obtain time dilation, no matter which frame you are in.
All observers are of course right, even if they donāt agree - but only in their own local frames. This is why measurements of time are not absolute, but depend on which frame they are performed in. This is quite a paradigm shift as compared to our own non-relativistic experience of the world, so it is quite understandable that it seems confusing or even paradoxical at first.
You might wonder whether there are quantities that are not frame-dependent, meaning all observers agree on them, irrespective of relative motion; the answer is yes, but to find them you need to account for both time and space simultaneously. Time dilation always goes hand-in-hand with length contraction, and vice versa. Note that what we are discussing here are kinematic effects - if you add gravity, things become more complicated still.
So the main points are:
1. Kinematic time dilation is a relationship between clocks (frames), and not something that āhappensā locally to a clock. Itās meaningless to say that a single clock is dilated. Nonetheless, this relationship is real (itās a geometric rotation in spacetime, as it turns out), and thus produces real physical consequences; itās not just on optical āillusionā based on what you might visually see (though of course optics are affected by this too, so there are corresponding visual effects).
2. Motion is also a relationship between frames, and not an absolute property of an object.
3. Measurements of time or space on their own are observer-dependent.
Hopefully this helps.