I have some points to discuss regarding your Figure 1. You did not state (probably due to constraints of the contest's essay length requirement) that when the light rays were initially emitted, both coordinate systems were coincident at their origins. In most renditions of this Galilean Transform diagram, the authors explicitly state that at coincidence of the origins, the "event" of the light source begins emitting the rays of light. This situation at the coincidence is the beginning of the epoch of time for the purposes of the argument. In other words, this is time tt=0=0. The emitter is obviously fixed to the S' coordinate system, (x', y', z') and "moves" with fixed coordinates, with the origin of that system.
You are correct to include the light rays in your diagram, most authors do not. In reality, the rays are actually just the same ray; both having been emitted simultaneously and in the same direction, at t=0.
Your depiction in figure 1 is actually a composite of three instances in time. The first instant is when the origins of both system S and system S' are coincident. The second instant in elapsed time is when the leading edge of the beam of light reaches the origin of S', and the third instant, of even longer elapsed time, is when the leading edge of the beam reaches the origin of S.
If we stipulate additional observers added to both coordinate systems, we can eliminate Einstein's problem with relativity of simultaneity. We do this by placing observer O't=0, and Ot=0, at the source; O't=1, Ot=1; at the origin of S', when the beam arrives at origin S'; and O't=2, Ot=2, when the leading edge of the beam gets to the origin of S.
At the first instant, (elapsed time = 0) when the origins of both system S and system S' are coincident, and the source emits the beam; observers at both origins do not see the beam because it has not traveled that far. Observers in both reference frames at the origin of S' (elapsed time =t1) see the leading edge of the beam arrive there. At yet a later elapsed time (elapsed time =t2), both observers at the origin of S see the leading edge of the beam arrive at origin S.
In the S coordinate system, at t0=0, the coordinates of (x,y,z) are identical in relation to the S origin, as the (x', y', z') coordinates are to the S' origin. So, in reality, those coordinates should be labeled: (xt0, ytt0, ztt0). At the later time, when S and S' have separated the time and distances = "vt1, and 2," the new "transformed" coordinates for (x,y,z) denoting the distances from the origins for each new position (t1 and t2) of x in S as the two reference frames recede from each other. (The coordinates for y, z, and y', z' remain at 0, since we are discussing the beam and not an expanding hemisphere of light as in the case of an isotropic emission as with a omnidirectional strobe light.
See my entry in this essay contest at A Logical Analysis of Albert Einstein's Mirror-Light-Clock Gedankin
Your comments will be appreciated. (I hope I have placed and named all the subscripts properly.