Hi Dave.
I do not quite understand what you mean by "interpretation of Time", but SR has no "interpretation of time"; it simply defines it (via the constancy of c and the resulting definition of simultaneity) in a logical way and then use that in a completely consistent manner. To me, it is a mystery why people have such difficulties to understand it.
Suppose you have never read the nonsensical notion of "moving clocks slow down" and just about spacetime structure and the fact that clocks (actually all objects) can have different paths through this structure and hence record different elapsed times between events; would that have made it easier for you to interpret?
In SR discussions it's not unusual to grant Players "Far Sight". The ability to watch each others clocks instantly and without Doppler considerations.
BTW, this "Far Sight" is just a network of clocks synchronized for each inertial frame of reference. Players don't have "far sight", but inertial frames have it through their set of synchronized observers. The
space-propertime view that I explained here makes use of this fact, although you may have noticed that I did not talk about simultaneity or time dilation here - just spacetime paths and proper time, the latter as read off clocks at close distance.
IMO, that's where the key to SR/GR lurks, with the other (more esoteric effects) just red herrings...