What is the specific derivation of the speed changes in the two equations from which Einstein originally inferred the relativity of simultaneity?

The two specific equations I’m asking about were stated at an introductory stage in the famous 1905 paper which launched Special Relativity. He didn’t explain how this derivation was arrived at; all I wish to understand is what his direct explanation would have been. The equations were:

$t_B – t_A = r_{AB}/(c – v)$

$t^*_A – t_B = r_{AB}/(c + v)$

These equations referred to a light ray emitted from $A$ at time $t_A$ and then reflected back from $B$ at time $t_B$, returning to $A$ at time $t^*_A$; where $A$ and $B$ are the ends of a rigid rod in a state of linear inertial motion, measured at velocity $v$ relative to its prior state of inertial motion. In this prior state its length was measured as $r_{AB}$. The times referred to were measured by observers located with clocks co-moving with locations $A$ and $B$ at velocity $v$; these clocks had been synchronised with those established as measuring the time on the rod prior to its state of motion changing to $v$. These events all occurred in empty space where the universal speed of light was assumed always to measure as $c$.

I ask my question about how $c – v$ and $c + v$ are derived for the equations above because, given Einstein’s two postulates, the following pair of equations would seem more appropriate:

$t_B – t_A = r_{AB}/c$

$t^*_A – t_B = r_{AB}/c$

The equivalence of these latter equations would of course have demanded a very different conclusion.