How does the relativity of simultaneity work in the classic train example?

[Jupiter]

I don't get how it works. Using the classic train example...
train with speed v rel. to S -->
[========C'=========>S'
A----------C-----------B----------S
|----L-----|

First of all, why do we use light to measure time? Why don't we use time to measure time, if that makes any sense? For instance, suppose at t=0, t'=0. Suppose A shines a light at t=0. Then doesn't S' measure that A shines the light at t'=0, ignoring this confusing light travel stuff? Same with B. What does it matter where the light goes and when it gets there?

Now if someone is sitting at C in S, and someone shines a light at A the time it takes to reach C is
t=L/c
But if someone is sitting at C' in S', he too will measure c for the speed of light. And according to this observer, the light has to travel the same distance L (half the length of the train), so the time it takes to reach C' according to an observer there is
t=L/c
So both these observers agree with the time at which the light was shone. Same thing with a light from B. So if something is simultaneous in S it should be simultaneous to S'.

Can someone point out my fatal flaw?

 

[Hurkyl]

Why don't we use time to measure time, if that makes any sense?

This is part of the point of learning all of these "paradoxes"; mentally we have some meaning of time that works for everyday situations, and it is difficult to accept that this mental meaning we have isn't the "right" meaning.

We can't use "time" to measure time, we use clocks to measure time. There are lots of ways to build clocks; but for the purposes of these thought experiments, light clocks by far have the simplest internals.

 

[Jupiter]

I still don't get how the light from A will get to C before C' if the speed of light is the same in each reference frame. In classical mechanics, S' would measure
v_light=c-v_train so the time it would take to reach C' would be
t'=L/(c-v_train)
Clearly, the larger v_train, the larger t' and only with v_train=0 will t'=t. But this is not classical mechanics. The speed of light does not change. Why won't both find
t=L/c=t'?

Hurkyl, what exactly is time?