What did Einstein mean in his 1905 paper when exposing the relativity of simultaneity? This question pertains to the section § 2. On the Relativity of Lengths and Times of Einstein's original 1905 paper "ON THE ELECTRODYNAMICS
OF MOVING BODIES".
I'm trying to figure out exactly what Einstein is saying in his demonstration that simultaneity is relative.  The conclusion is very familiar to me, so I'm not asking for an alternative demonstration.  I want to understand the passage quoted below; and, in particular, the footnote.
Apparently he is saying that when the system at rest measures the length of the rod by "simultaneously" recording the positions of the ends, each moving observer is to set his clock to match the rest-frame time of the measurement event at his location.
Then it gets weird.  He seems to be saying that the clocks moving with $\rm A$ and $\rm B$ are to continue to match the coinciding rest-frame clocks as they pass them, into the future.  But for that to happen, the moving clocks will need a different unit of time than clocks at rest.
Furthermore, for my interpretation to be correct, the statement "[T]hese observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks." will have to mean that the moving observers are not setting their clocks by this method, but are merely checking to see if they are synchronized; and those moving clocks are "slaved" to local rest-frame time.  I say this because the footnote indicates that both the rest-frame and moving clock at, say the reflection event, are to read time $t_{\rm B}.$
Am I reading this correctly?

We imagine further that at the two ends $\rm A$ and $\rm B$ of the rod, clocks are placed which synchronize with the clocks of the stationary system, that is to say that their indications correspond at any instant to the “time of the stationary system” at the places where they happen to be. These clocks are therefore “synchronous in the stationary system.”


We imagine further that with each clock there is a moving observer, and that these observers apply to both clocks the criterion established in § 1 for the synchronization of two clocks. Let a ray of light depart from $\rm A$ at the time [footnote]  $t_{\rm A}$, let it be reflected at $\rm B$ at the time $t_{\rm B}$, and reach $\rm A$ again at the time $t^\prime_{\rm A}$. Taking into consideration the principle of the constancy of the velocity of light we find that


$$t_{\rm B}-t_{\rm A}=\frac{r_{\rm AB}}{c-v} \text{ and } 
t^\prime_{\rm A}-t_{\rm B}=\frac{r_{\rm AB}}{c+v}$$


where $r_{\rm AB}$ denotes the length of the moving rod—measured in the stationary system. Observers moving with the moving rod would thus find that the two clocks were not synchronous, while observers in the stationary system would declare the clocks to be synchronous.

Footnote:

“Time” here denotes “time of the stationary system” and also “position of hands of the moving clock situated at the place under discussion.”

 A: 
Then it gets weird.

Yes, that is an apt description. Remember, this was brand new. There weren’t any of the standard pedagogical techniques then and he couldn’t ask for any help in making a better explanation. So this specific section was a little weird and no subsequent author (including himself) ever used this argument again.

Then it gets weird. He seems to be saying that the clocks moving with A and B are to continue to match the coinciding rest-frame clocks as they pass them, into the future. But for that to happen, the moving clocks will need a different unit of time than clocks at rest

Yes, that is correct. A similar thing is actually done with GPS satellites. They are moving in the earth centered inertial frame (ECIF), and so their clocks are adjusted so that they do not keep correct proper time but rather they match the ECIF time.

the moving observers are not setting their clocks by this method, but are merely checking to see if they are synchronized

Yes, you are reading it correctly.
A: The following is a comment; I'm posting it here because comment space is too small for this message.
(I endorse that the comment space is small.)
In the preceding paragraph, 'definition of simultaneity', Einstein presents the synchronization procedure that would later be named 'Einstein synchronization procedure'. Then Einstein wrote about that procedure:

We assume that this definition of synchronism is free from contradictions

Well, for sure: in Minkowski spacetime the Einstein synchronization procedure does not give rise to self-contradiction.
However, it is interesting to consider how Einstein sychronization procedure fares in newtonian space&time
Let me discuss the case of propagation of sound.
Take a train, with three emitters/receivers of sound.
Run the synchronizatio procedure with a range of different velocities of the train, and compare the results.
The procedure:
The emitter in the middle sends pulses of sound to the left and right. As each pulse is detected at the end of the train a sound pulse is sent back to the middle.
As we know: for each velocity of the train the two reply-pulses arrive at the same time back at the middle of the train.
However, the duration of a cycle is not the same in each case. The duration of a cycle is shortest when the train has no velocity with respect to the air through which the sound pulses are propagating.

(In the two images below the yellow lines are hardly visible, my apologies for that. The images still convey the message, though.)

The diagram above shows a plot of the sound pulses (yellow) as they travel throught the air, for the case when the train is stationary with respect to the air. The red lines plot the motion of the three emitters/detectors on the train.


The diagram above shows a plot for the case when the train has a velocity relative to the air.
What this shows that if your apply the Einstein synchronizatio procedure in newtonian space&time, trying a range of velocities, then you can home in on the velocity where you are stationary with respect to the medium through which the waves are propagating.

In Minkowski spacetime however, you get a very different result.
When the motion is taking place in Minkowski spacetime then you have to take time dilation effects into account.
As we know: if you do the Einstein synchronization procedure in Minkowski spacetime, using pulses of light, and you do that for a range of velocities, then a co-moving clock will report the same duration for the cycle for every velocity.

Returning to the assertion by Einstein:

We assume that this definition of synchronism is free from contradictions

In retrospect we see that to assume that no self-contradiction will arise is already sufficient to necessitate Minkowski spacetime, instead of Newtonian space&time

Anyway, what is most striking to me is the tone of voice of Einstein in the 1905 article. I assume he was fully aware how profound his proposal was. But his demeanour is casual.
Of course, in a scientific article it is standard to use a neutral voice. But still.
I like to think Einstein was being pragmatic here. Einstein did not state upfront: "This will change all of physics!". Instead Einstein presented the concept piece by piece, and at no point does he let on that he is ushering in a revolution.
