Alice notices Bob drifting past with his clock ticking away. What makes the clock Bob’s is that it moves along with him: it records
Bob-time. The universal speed is the universal speed, but Alice notices that the pulse for Bob‘s clock has to cover a greater distance between collisions with the mirror.
Now you can compare the different rates of ticking as a result of movement.
The light paths suggest that there should be different rates of ticking, based on the universal speed and the various distances covered by the light pulse between the mirrors. That’s why all
moving clocks go slow, and so why time passes at different rates for any clock that is not moving with you. (The ticking of
Bob-time agree when Alice and Bob don’t move apart or move together.) These changes will not affect the light cones drawn just for the owner of a co-moving clock. But these light cones show how personal time is: your record of the clock's ticking depends on how fast the clock is moving past you: watchers record different durations from witnesses. Because clocks can also measure distance, how fast the clock is moving past you also affects your records of distance. Records of both space and time or space-time are affected by relative motion.
There is nothing special about being a Bob or an Alice; Alice notices that Bob‘s clock is running slow, and Bob notices that Alice’s clock is running slow. Whoever you are, the fastest clock is
you-time, shown on the clock that travels with you. No one notices you age faster than you do, which might be a relief.
Choose one point of view or another. You have to choose a point of view of one local witness, although it might not be one of this pair.