". . . a ray of light proceeds from A at time tA towards B, arrives and is reflected from B at time tB and returns to A at time tA'. According to the definition both clocks are synchronous if tB - tA = tA'- tB. We assume that this definition of synchronism is possible without invoking any inconsistency. . ." (Einstein 1905, my translation)
The equation, in this citation, expresses the fact that the time in which the light travels from A to B is equal to the time required for the return path. If that equation defines what we mean by synchronized clocks then it follows, from elementary logic, that if these times are not equal, the clocks are not synchronized. That was how Einstein came to the conclusion that clocks in motion, where the two paths are unequal in duration, cannot be synchronized, that they do not tick at the same rate. The logical inference is correct, but that equation cannot be the definition of synchronization. So if the premise is wrong, the conclusion is also wrong. He apparently does not realize that unless the clocks are synchronized in advance, you can't do the subtraction on either side. Subtraction only makes sense if you are dealing with the same time-piece, or two identical, synchronized clocks.
If the clocks are synchronized and there is no motion, the equation is true, but not vice versa. The error in logic is analogous to defining "crime" as the act of stealing, and then concluding that since you didn't steal, you did not commit any crime.
Furthermore, you can't write tB - tA without considering who knows this. You therefore need (a) previously synchronized clocks at A and B, because otherwise the subtraction makes no sense; and (b) you do the subtraction (tB - tA) as the observer at A, at a later time, when you find out what was the reading of the clock at B when the light pulse arrived. At that point, you can also know (tA' - tB), the time it took the light on the return path to A, and can decide whether they are equal, in which case there was no movement between A and B while the light was going back and forth.
Einstein's argument amounts to using synchronized clocks to prove that they were not synchronized, a contradiction, or whatever else you may call this logical fallacy.
One counter example, that demonstrates that two bodies in relative motion can have synchronized clocks, is sufficient to explode the contention of Einstein and others that time is local and flows differently, and that clocks on two such bodies cannot be synchronized.
Here is where the Doppler effect plays the critical role. When two bodies approach each other and reach a minimum distance, after which they separate, the Doppler effect will reverse at the moment of closest approach. (We can use monochromatic light and observe red and blue shifts, or discrete pulses and measure the change in their spacing.) That is true for observers on both bodies, and the two clocks can be set to, say, 1 o'clock at that moment — which will be the same moment for both bodies without the need to communicate the event. (The concept 'closest' does not pertain to just one body. If A is closest to B, at a given moment, then B is also closest to A.) Similarly when two bodies move away from each other, reach a maximum separation and then move towards each other, the Doppler effect again reverses at the moment of maximum separation - and it will be the same moment for both bodies without a need to communicate. Physics alone is needed to see this. So now we have two moments in time when the clocks agree. The second moment can be given the name 2 o'clock. Whether it is earth and moon, or Mars and Venus, as long as we can ascertain that the period remains the same over 'time' we have well synchronized clocks for two bodies in relative motion.
As regards simultaneity: in Einstein's 1917 book, his attempt to justify the relativity of time rests on the analysis of the concept of simultaneity. Einstein begins this book by questioning the traditional concepts of time and space, and asserting that these traditional ideas are in conflict with the idea that light propagates at a constant speed. He maintains that it is the relativity of "simultaneity" which leads us to a different view of reality, and to the concept of the relativity of time and space, as opposed to the absolute nature of time and space. (This book, written 12 years after his original paper, is both clearer and more clearly wrong than his original work. The earlier work dove into mathematics very quickly and so obscured the reasoning that led him to his equations.)
But Einstein does not realize that there are two distinct and independent types of "simultaneity." One type occurs when a single observer is aware of two distinct signals simultaneously, for example, hearing a doorbell and a siren at the same time. We can call this e-simultaneity, i.e. one observer, two events. The other type occurs when two observers become aware of one event at the same time, such as an explosion or an earthquake. We can call this o-simultaneity - two observers, one event.
"Relativity" is true only for e-simultaneity, and means that different observers, because of their different locations, may not agree that two events or signals are simultaneous (event, or e-simultaneity). This type of simultaneity does not depend on the use of a clock. Each observer simply needs to notice when two events are concurrent - and two observers may disagree, because of their different location with respect to the two events.
But it is the other type of simultaneity that is needed for his theory. In that case, the simultaneity is determined by the clock time, which must be the same for both observers, (o-simultaneity). They can infer that they noticed the event at the same time by comparing the readings of their watches - assuming these were synchronized in advance. The idea of relativity does not apply to this type of simultaneity.
Einstein confuses these two types of simultaneity, and so comes up with the bold assertion that simultaneity is relative. What he needs is for o-simultaneity to be relative, but borrowing the idea from e-simultaneity is not legitimate.
In addition, Einstein is unclear when he refers to the simultaneity of events. Sometimes he means the origin of signals, and sometimes he means the signals as received by one or two observers.
We can use a bolt of lightning and an exploding balloon or dirigible as an illustration of the difference between e-simultaneity and o-simultaneity. In the limiting case, which we could call true simultaneity, the bolt of lightning strikes the dirigible and it "simultaneously" explodes. Now all observers, wherever they are, or however fast they are moving, will see these two events "at essentially the same time." They originate at about the same time from the same place, and therefore take the same time to reach any given observer (e-simultaneity).
However, observers may be at different distances from this double event, so they will not experience it "at the same time." We are using the phrase "at the same time" in two different ways. It can mean "concurrently," if it relates to the two events, but it also can refer to the clock time when we compare the time at which the signal reaches different observers (o-simultaneity).
By not keeping track of the different meanings, we can easily draw false conclusions. e-simultaneity does not imply o-simultaneity and the relativity of one does not imply the relativity of the other.
For Einstein's theory, it is o-simultaneity that is the real issue, whether two observers in relative motion can compare experiences in a time frame common to both. He needs the relativity of e-simultaneity, but it does not apply to o-simultaneity, We see that Einstein's attempt to justify the consequences of SRT by appealing to the concepts of simultaneity and synchronization, relies on ambiguities and contradictions in language and logic. His appeal to common sense, to support his faulty mathematics, does not strengthen the case for relative space and relative time - it weakens it.