Trying to explain non-simultaneity

Suppose we are standing 2 light-seconds (about 600000 km) apart, and not moving with respect to each other. There is a third entity precisely between us, also not moving with respect to us. We'll call our shared reference frame the Stationary Reference Frame, or SFR for short.

The third entity sends out a pulse of light to each of us. Each of us sends out a pulse of light towards the other upon receiving the first pulse.

These second pulses pass the middle entity at the same instant, 2 seconds after it sent the initial pulses.
An observer sitting in the middle would see both of us flashing a light at the same time.

Another second later, my pulse reaches you and vice versa. Each of us receives the other's pulse 2 seconds after sending their own. Since we know the distance between us (2 lightseconds) and the speed of light, we know the time it took the light to travel that distance: 2 seconds.

All good so far. Now let's rewind to the beginning and add a *moving* observer. I'll say he's moving at two-thirds of lightspeed, starting at your position and moving towards me. The Lorentz factor in this situation is roughly γ = 1.34.

The moving observer encounters the initial pulse (from the middle to you) when he's 2/5 of the way from you to the middle. In the SFR, this occurs 0.6 seconds after firing the initial pulse.

0.4 seconds later (in the SFR), we both receive the initial pulse and send out our own. The moving observer is now 2/3 of the way from you to the center.

0.8 seconds after that (in the SFR), the moving observer encounters my pulse. He is now 1/5 of the way from the center to me.

Another 1.2 seconds later (in the SFR), your pulse catches up to the moving observer when it is exactly at my location.