In the textbook version it is easy to see how the path length increases in accordance with Pythagoras's theorem. With the light travelling parallel to the direction of motion, the pulse will travel further in order to get from A to B than from B to A, but the total path length over a complete cycle remains the same regardless of the relative speed. So the clock runs at the same rate, whatever the relative speed. But hang on: the Fitzgerald-Lorenz contraction will shorten the distance between the mirrors by a factor of gamma. So the clock will run faster with increasing relative speed. :-)

In the textbook version it is easy to see how the path length increases in accordance with Pythagoras' theorem. With the light travelling parallel to the direction of motion, the pulse will travel further in order to get from $A$ to $B$ than from $B$ to $A$, but the total path length over a complete cycle remains the same regardless of the relative speed. So the clock runs at the same rate, whatever the relative speed. But hang on: the Fitzgerald-Lorenz contraction will shorten the distance between the mirrors by a factor of $\gamma$. So the clock will run faster with increasing relative speed.