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The way to deal with virtually all problems of this
ilk is to DRAW THE SPACETIME DIAGRAM.
...
> The proper time interval between two
> events in a given reference frame must be measured with one clock only,
Not true.
The conventional wisdom is diametrically opposite.
You want a "lattice of rods and clocks" in each
frame. (The clocks are of course synchronized in
that frame.) This allows the observers in that
frame to measure the time of an event _at_ that
event, using the colocated clock and no other.
...
> Your setting is more complicated (and
> therefore probably unsuitable as a thought experiment designed to clarify
> the situation)
More complicated than what?
The stipulated setting isn't complicated in any
absolute sense, and is 100% suitable as a thought
experiment.
There is however no notion of proper time involved.
The proper time for a photon to go from point A to
point B is zero, if the concept can be defined at all.