Try to imagine that no movement occurs. That is, both depictions are mathematically static. That they are simply given as two different mathematical scenarios.

Then neither one has anything to do with the other. If S = S' here, and S = S' - D here, and they are two different scenarios..

Then x = x' in the first scenario, x = x' - D in the second scenario, and neither one have anything to do with each other. Scenario 1 x (hereafter x

_{1}) and Scenario 2 x (hereafter x

_{2}) are not the same

x

_{1} = x'

_{1} and x

_{2} = x'

_{2} - D

x

_{1} =/= x

_{2} because.. as you said, they are two different scenarios and as such, have no relation to each other whatsoever.

The problem is in deciding what x means in the equation x' = x-vt or more specifically, x' = x-D.

Not really? x means any point on the x axis. Since we're talking about two different x axes, we need a way to distinguish them and you're using the ' mark, so that's what I'm using. x is any point on the x axis, and x' is any point on the x' axis.

In general, "given coordinates remain fixed FOREVER wrt their own system's origin".

I.. think I agree with that, as that's the whole point. With an origin of 0,0 (or 0,0,0, or 0,0,0,0.. you get the idea) the entire existence of something like 2,1 depends utterly on 0,0 existing 2 and 1 point away on the respective X and Y axis. So no, 2,1 cannot move with respect to 0,0

Now... that being said...

If 0,0 and 0,0 are the same on two coordinate systems, and 1,1 and 1,1 are the same, and -1,-1 and -1',-1' are the same and both coordinate systems are described as planes... then they're coincident...

I would argue that we cannot directly say that 50,587 = 50',587' because they are on two different systems. Rather, we say that 50,587 is 50 and 587 units away from 0,0 and that 50', 587' is 50 and 587 units away from 0',0', and that 0,0 = 0',0' | 1,1 = 1'1', | -1',-1' = -1,-1, so we can safely operate as though 50','587' and 50,587 occupy the same space. We.. uh, just shorthand that as a given, but if we had to prove it, that's how we'd prove it. That the three points given on the first system (1,1 , 0,0 and -1,-1) are described as coincident with the defined points on the second (1',1', 0',0', and -1',-1') and the two coordinate systems are defined as planes.

Now once 0,0 and 0',0' no longer occupy the same space, any existing relationship we assumed between the two systems goes completely out the window and we have to rebuild our system of understanding. If at one point 0,0 and 0',0' were the same, our rule has to allow for that... but it also has to allow for 15,68 = 0',0' or whatever. So we come up with some formula and can draw relationships between various points of x,y and x',y' based on that formula.

Which, if I'm not mistaken, is the vd thing that keeps being mentioned. As I said, I've got high school level math skills that have been rusting like hell over the last decade, so I doubt I'm using anything close to the right language and hopefully didn't screw up too many things.

One can observe that the x abscissa lengths and the x' abscissa lengths are not altered when the S and S' systems appear in the second depiction wrt their abscissa lengths in the first depiction.

That.. sounds like a really fancy way of saying "On S, X is equal to X in both depictions such that this particular point is still 3 units away from X, and on S', X' is equal to X' in both depictions such that this particular point is still 3 units away from X' " to which I can only say... "... yes? That's the entire point of having an axis? That points defined on it remain a constant distance from their 0 point?"

The only difference between the two is that S and S' were on the same point in space (or as close as we can physically get with the model, but was defined as being overlapping and Coincident) and now S' is no longer sharing space with S but is now instead... if I had to guess, I'd say that S' is now on 3. So 3,0 = 0',0'

Which, of course, now means that 3,0 and 3',0' are no longer in the same space. Rather, x,y = x'-3,y'

Because S(0,0) = S'(0-3,0)

or something. Not sure what the proper way to write that would be. See above re: High School Level Math Skill (Rusty)