[TyroJack (OP)]

When I think of time as the 4th dimension I imagine it broadly applying to the Universe which ages uniformly -- 3.8 billion years or so -- but how does that work if different observers, in relative motion, measure time differently?

A dimension gives us the ability to place something with a coordinate on an axis and points on an axis are in a strict order.

So how can time be a grand 4th dimension applying universally in Spacetime?

It seems much easier to see it as the 4th dimension in a particular frame of reference, assigning a time coordinate to objects or events within that frame.

[alancalverd]

The introduction of "relative motion" sows the seeds of confusion.

Suppose Alf, Bill and Charlie stand in a straight line in alphabetical order. From Alf's point of view, Bill is closer than Charlie. But now let Alf move along an arc to end up at the other end of the line. En route, B initially appears closer than C, then equidistant, then further away.

You have put your finger on the nub of relativity. There are no absolute coordinates.

[Colin2B]

Suppose Alf, Bill and Charlie stand in a straight line ....

 ......You have put your finger on the nub of relativity. There are no absolute coordinates.

Excellent example of why it’s not just time that is relative.

[yor_on]

It's not as weird as you think. There is something called 'SpaceTme intervals'

That one will help.

the other thing to remember is that all other clocks you define goes out from yours.
And so theirs.

That's actually a 'constant'.
What it does is to give us a logic

[Halc]

So how can time be a grand 4th dimension applying universally in Spacetime?

It seems much easier to see it as the 4th dimension in a particular frame of reference, assigning a time coordinate to objects or events within that frame.

Exactly so. That's just what a frame of reference is. It is the assignment of a coordinate system to spacetime.
There is no X axis that applies universally in spacetime.  You can assign it any direction you want, and ditto for Y.  It's nice if they're assigned orthogonal, else the mathematics comes out more difficult.
Likewise, selection of an inertial frame of reference is the same as what you're doing with X.  You're choosing a direction for it to be aligned, and doing so orders all events. All other axes can be assigned orthogonal to that, but there is no universal direction for the time axis.

[CPT ArkAngel]

The causal order of events is still the same for all observers.

[TyroJack (OP)]

Thank you, one and all.