A discussion on this question on 'grounding absolute time' prompted the following possibly clarifying, possibly unclarifying thought:
In Newtons Principia, he defined two concepts of objective time:
absolute, true and mathematical time, of itself, and from its own nature, flows equably in itself and without regard to anything external.
And he adds:
relative, apparent and common time, is some sensible and external measure of duration by means of motion
How do these two concepts alter in SR; if in fact they do, which post-hoc, we know they must?
Now the concept of relative time must change in SR, because it is explicitly, as Newton defines it, dependent on the concept of motion; and it is this concept that is altered in SR: ie the motion of light is not Galilean - being always constant.
And an argument first presented by a Einstein (using the motion of light as a clock - this is his version of Galileos pendulum) shows that in fact this is the case.
I would suggest, however, that the absolute concept does not alter, as it does not refer to anything external; thus how a particle moves in space does not affect this concept of time; thus proper time in SR ought to be that of absolute time in Classical Mechanics.
But is there a good heuristic argument that shows that this is in fact the case?
This a problematic since only when two particles are at the same place can we directly compare their proper times, and see that they flow at the same rate; when separated we cannot compare directly.
The following argument suggests itself:
Consider two particles at the same place in space; they then move away from each other in opposite directions at the same speed and in a straight line.
Taking an inertial frame with particle A, we see that the clock held by B has slowed; analogously when taking an inertial frame with particle B, the clock held by A slows; these are in fact symmetric situations - and though we cannot compare the proper times of A and B directly, we see that by symmetry that they must flow at the same rate.
But this was a special situation; what if, say the two particles are moving with uniform but arbitrary linear speeds? Then, I would suggest that there is an inertial frame that resolves this situation to the situation above.
Thus proper time, I would suggest is absolute time; even though the concepts objectively belong to two different physical theories.
Further, one can consider GR; recalling the elevator Gedanken-experiment which free-falls; and when within and without probing externally - ie there are no windows - one cannot tell whether one is within a free-falling elevator or not.
Again, recalling Newtons definition of absolute time, which flows without reference to anything externally, we see that the proper time within the elevator can be legitimately called absolute time - even if it is not.
The concept that connects all these situations, I would suggest, is that all inertial frames everywhere and at all times are the same - so proper time again can be legitimately connected with absolute time.
Finally is this usefully correct or unusably and unusefully uncorrect?