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Maybe I can show the way I think it should be by using Galilean
relativity as an analogy. I know that the Galilean transformations
aren't used to find the spacetime interval between events, but I think
they can be used to make my point.
Say the situation is the same as before, so that event E1 occurs at the
coincidence of the origins of F and F'. Then at time t = t, event E2
occurs at x = vt. The Galilean transformations give the coordinates of
E2 in F', in this situation, as
x' = x - vt = vt - vt = 0
y' = y = 0
z' = z = 0
t' = t
Now suppose that observers in F' are asked to give the distance between
E1 and E2. Should they say that x' = 0 (which is what the analogous
Lorentz transformations say), or should they say x' = x = vt = vt'? I
say they should say the latter.