Disclaimer: I did not study physics beyond classical physics (apart from some basics of quantum mechanics), so I apologize if this question is ill-posed.
My question is more clear if you see the snapshot at 5:00 of this video (it is not necessary to see the whole video).
In that snapshot, call A (Alice) the photon-clock at rest and B the other. It is clear from the spacetime diagram that B (Bob) ticks slower (according to the frame of Alice). Note that sufficient time has been allowed to pass in order for the ticks on the photon-clocks to be different (you can imagine any arbitrarily big difference in ticks; in the snapshot, it is of only one tick, but we can make it arbitrary). Imagine, now, the following modification of the experiment.
Both photon-clocks have a digital counter of "ticks". A flashlight (F), synchronized with Alice, fires a photon exactly when clock A starts to tick. The photon from F satisfies the following condition: when Alice clock is at tick 4, the photon from F must meet clock B. The spatial position of the flashlight in order for this to happen can easily be found via a simple drawing (by using the horizontal line of Tick 4 and a -45° line intersecting the horizontal line and the clock B). When the photon from F reaches B, the digital counter stops updating. Alice also stops updating its digital counter at exactly 4 ticks.
Alice and Bob can now compare the numbers in their digital counters. Alice will find 4 ticks and (if you watch the snapshot) Bob will find 3 ticks in his digital counter, since the photon from F stopped the digital counter from updating. Thus, moving clocks tick slower, indeed.
Up to now, everything appears fine, but now I see a problem.
This problem has been solved using Alice's frame (which is the same frame of the Flashlight, since both are at rest). From the frame of Bob, however, both Alice and F are moving (at constant speed), while B itself is stationary, so Alice should have less ticks on her clock.
But the experiment was setup for Alice to notice a difference in ticks: sufficient time was allowed to pass. So the photon from F cannot have arrived when both photon-clocks displayed the same digital number (which happens, for example, before the very first tick).
So, what does this mean? Is it correct? Does Bob realize that his clock was ticking slower than Alice?
In the two clocks experiment (with no flashlight), Alice thinks Bob's clock is slower, but Bob, symmetrically, thinks Alice's clock is actually slower. This is, essentially, due to relativity of motion, since there is no such a thing as a "privileged" inertial frame. But, now, only numbers of ticks (say, $n_A$ and $n_B$) are compared, so either $n_A > n_B$ or $n_B > n_A$ ($n_A = n_B$ was ruled out by careful positioning of the flashlight).
