Time dilation: speed relative to what?

[Dale]

from the embankment, where the two events are simultaneous, the interval between the two lightning strikes would be s2 = 0 + x2 + y2 + z2

OK

in the trains frame it would be -(cγt')2 + (γx')2 + (y')2 + (z')2

No, in the train's frame it is $s^2 = -c^2 t'^2 + x'^2 + y'^2 + z'^2$. It most definitely does not equal what you have written.

but as y' = y and z' = z this reduces to
x2 = s2 = -(cγt')2 + (γx')2

In the train's frame $ct'=-vx \gamma/c$, $x'=\gamma x$, $y'=y$, and $z'=z$.

So $s^2 = -c^2 t'^2 + x'^2 + y'^2 + z'^2 = - v^2 x^2 \gamma^2/c^2 + x^2 \gamma^2 + y^2 + z^2 = x^2 + y^2 + z^2$

 

[Grimble]

Regretfully I'm not sure to understand the meaning of those sentences. Perhaps you are thinking of a train in rest in one inertial reference system, compared with a train in rest in another inertial reference system? Indeed the physics should be the same.

Yes indeed I apolodise for my wording being a little slack there.