I'm not very good at relativity and frames of reference, so perhaps I'm not best to reply to this one, but the initial sentences sounded a bit funky to me:
A fundamental concept of Special Relativity is relative motion. Assume two inertial systems S and S' are moving in completely arbitrary paths. The instinct is to say that the relative motion is 3dimensional. One could not be faulted for that view, but relative motion is a 1dimensional effect with two degrees of freedom.

If the two reference frames are inertial, that means that the relative motion between them is not completely arbitrary... Perhaps the OP should replace "arbitrary" with "arbitrary rectilinear motion".
Then comes this bit...
Draw a straight line segment between S and S' and call that distance d. There are many paths the systems can take where d is invariant. There are other paths where d is variable. It is this variation of d that is the essence of relative motion, it occurs in a single dimension, the line segment connecting the two origins.
Within that dimension the translation of S and S' is one degree of freedom, the rotation of S and S' is the second degree of freedom.

I think it's correct, but is it important to think of the system in terms of degrees of freedom required to explain the relative motion of the frames? I would say that it's more important to consider whether the frames are inertial or noninertial in terms of acceleration (for example). Consider two cases:
1. Two inertial frames, S is stationary, S' has a constant velocity of 1 m/s to the right relative to S.
2. One inertial frame and one noninertial frame; S is stationary, S' is initially stationary, but then accelerates to the right at 1 m/s^2
In the first case, the line connecting two stationary points in the frames is determined by a single vector (one degree of freedom) with increasing size. This is also true for case 2, but now the laws of physics in S' requires fictitious forces in order to fully characterise the motion within that frame. You could argue that the proper "second degree of freedom" is the acceleration, but that's one way of determining whether a frame is inertial or not anyway...