Like most people I struggle to get my head round special relativity. I read a superficially-plausible explanation for why a moving clock runs slow from the point of view of an observer in a rest frame - this is the light clock that bounces light perpendicularly to the direction of travel between two mirrors: because the mirrors are moving, from the point of view of the rest frame, but not the moving frame, the light path is longer and each 'tick' takes longer. But what happens if the light is bouncing parallel to the direction of travel? If the mirrors were sufficiently far apart to measure it, would the rest-frame observer see alternating long ticks and short ticks? Surely not, but what am I missing??
If there was a difference in the transverse and longitudinal light-clock, then large Michelson interferometer would be able to detect the direction of the Earth's motion.
They can't. Just ask the LIGO guys.
So why do all the explanation use the transverse version? Because it is easy, and it doesn't require that you already know about length contraction.
The use of a light clock, in particular, is attractive because it couples directly to the speed of light postulate: the distance traveled and the (invariant) speed of light fix the time it took
There are other ways to introduce the subject which avoid exhibiting that particular construct. Tatsu Tacheuchi's little book is a brilliant example.