Let's discuss one aspect of the diagrams at a time, so that we can identify each problem separately.
In your diagram on the left, Einstein's embankment frame, you show that the embankment observer thinks light traveled .5 seconds to travel from the source at the midpoint of the train to the receiver attached to the side of the train, which is .5 light seconds from the source.
James, that is incorrect. The embankment observer will see that the receivers on the sides of the train have moved to the right in the diagram, as the receivers on the sides of the train are affixed to the train, and the train has traveled to the right in your diagram. So, in the bottom pic of your left diagram, the light has not yet reached the side receivers at the midpoint of the length of the sides of the train. In other words, in Einstein's embankment frame, the light can't possibly reach the side receivers in .5 seconds, because light has to travel a greater distance than .5 light seconds to reach them. The radius lines of the light sphere from the center of the sphere to the point the receivers are attached at the midpoint of the length of the train will be at an angle to the right, which means the radius lines are longer than .5 light seconds in length.
Please explain that before we continue.
I've attached a pic to show the distance the light has to travel to reach the side receivers, of which the light has clearly not reached yet. Also note, that if the light sphere needs more time to expand to reach the side receiver, the train is also traveling to the right, so the time is even greater, as the angle increases. The light sphere will grow outside the train's front before the light reaches the sides.