Quote by gravuniverse
I can only use inertial observers as stated in the postulates in order to measure the speed of light the same in all inertial reference frames and then to perform the mathematics accordingly.

But "inertial observers" doesn't necessarily imply the Lorentz transformation unless you assume
both postulates of SR. Inertial observers as defined in Newtonian physics all observe the same laws of physics (first postulate satisfied), and all see each other traveling at constant velocity, but there's no invariant speed postulate and their coordinates transform according to the Galilei transformation. Likewise, I showed that if you just wanted to satisfy the second postulate but not the first, you could have a family of coordinate systems that all see light moving at c, and that all see each other traveling at constant velocity, but where the coordinates transform according to a different transformation. If you're going to go mucking about with the postulates, you can't
start out assuming that the phrase "inertial observer" will mean exactly the same thing as it does in SR, with different frames related by the Lorentz transformation, that'd just be circular reasoning rather than an actual "derivation'.
Quote by gravuniverse
Okay, well if one messes with the distance coordinization in order to make the ticking working out the same

It's the time coordinate that determines the rate of ticking, not the distance coordinate.
Quote by gravuniverse
in the reality of the noninertial observer

Again, "the reality of the noninertial observer" is meaningless since there is no single way to construct a coordinate system where a noninertial observer is at rest. You have to talk about coordinate systems, not "observers".
Quote by gravuniverse
but then one would have to design a different coordinate system for accelerating away from the rest frame as accelerating back, otherwise how would something like the twin paradox work out?

No, you'd have a single noninertial coordinate system, not two different ones for different parts of the trip. Since time dilation doesn't work the same way in noninertial coordinate systems as it does in inertial ones, there's no problem getting the twin paradox to work out, at some point the inertial twin would just have his clock ticking faster relative to coordinate time than the noninertial one.
This section of the
twin paradox FAQ features a diagram showing what lines of simultaneity could look like in a single noninertial coordinate system (drawn relative to the space and time axis of the inertial frame where the inertial twin is at rest):
You can see that during the phase where the noninertial twin "Stella" is accelerating, the clock of the inertial twin "Terence" will elapse much more time than hers. Lines of constant position in this noninertial system aren't drawn in, you could draw them any way you like (including curved lines so that Stella could be at a constant position throughout her trip) and have a valid noninertial system.
Quote by gravuniverse
In any case, I am only considering inertial observers with each having the same coordinate system to keep things simple.
Right, I editted my post since my last reply, sorry. The trailing observer having a greater acceleration must be what I was thinking. Anyway, I can now see that what is occuring with the light catching up to the accelerating observer can all be worked out from the frame of an inertial observer as you said.

And you understand how in Rindler coordinates, any given clock in that family of accelerating clocks can be ticking at a constant rate relative to coordinate time, and occupying a fixed coordinate position?