Things may be stored differently from the way I experience them

It's two points, one in each of the 2D spaces, because you can't allow:

point A at 1, 2 and 3, 4
point B at 5, 6 and 3, 4

If you do, then two points are sharing the same position on the second surface.

But two different points in four-dimensional spacetime can share the same position on the time axis, and still be different points in spacetime.

Imagine "Flatland" rewritten by J. W. Dunne, so you'd have two dimensions of space and two of time. (Actually, I think Dunne had infinitely many dimensions of time, but just think about two.) Then that space-time is four-dimensional (2+2) just like conventional (3+1) spacetime.

But two different points in four-dimensional spacetime can share the same position on the time axis, and still be different points in spacetime.

That's because it's a 4-dimensional continuum, so the two points are in different places. But with two 2-dimensional surfaces, things get confusing.

Is each point really a pair of points, one on each surface? You might be able to work it out if you think of it this way.

But you said "... here I'll discuss information as if it were stored in just 2 dimensions." That suggests that you really mean information to require only two coordinates.

Anyhoo, I'm beating this up more than it needs to be.

~~ Paul