You must first understand that SR is an insane theory. It says t1=T,
t2=T but t1 is not equal to t2. Those who accepted this theory were
hardened criminals. Holocaust had no effect on them. They were there
to promote a mad man as great physicist. Projections, mathematics,
related equations are meaningless unless they are not applicable to
practical setup. So we arrive at the immediate paradox. If a particle
travels in the space of frame S and since space of S never contracted,
how can a particle travel a contracted space? But you cannot stop
donkeys braying. That is their habit. It is unfortunate that physics
is full of some donkeys who can’t understand simple things and some
criminals who do not admit inconsistencies. By the way explanation of
projection is correct. This cannot be wrong since mathematics cannot
be wrong. Mathematics and geometry is incapable of understanding
nature of application. We face problem only when we try to fit this
projection on the length of actual path.
You are not explicit about your second statement. Possibly you want to
say the same as stated above. If all clocks tick at the same rate in
their respective frames, then this is possible only when time is
absolute. Therefore measurement made from the other frame is wrong and
we get this wrong measurement because theory of relativity is wrong.