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I think the Galilean example helps avoid confusion. Do I understand
correctly, then, that you are simply interested in the distance between the
points where events E1and E2 occur?
Do I further understand correctly that
you would like that distance to be the same in both reference frames?
If my understanding is correct, then please explain why you seem to choose
to give priority to frame F, by wanting the distance measured in F' to not
be zero. That seems to violate the basic principle of relativity, either
Galilean or special. Maybe the stake driven in at event E1 has moved
according to F', but the event of the stake-driving does not "move". The
distance between the stakes at the time of E2 is not given by transforming
E1, since E1 is an event that occurs at a specific time (zero). If you
want the distance between the stakes at the time of E2, you need an event
E3 that occurs at the location of the first stake at that time, and you
must determine that time.