Obviously there's some conflict between everyday usages of these words and their mathematical definitions. I agree with the OP about people living on a sheet of paper not being able to tell if it had been rolled or not.
This caught me eye though:
Please correct me if I'm wrong, but isn't this exactly
the kind of thing the curvature of space-time effects? If you have a massive object, space-time curves around it. Think about if you were surveying distances by how long it took you to walk from place to place, moving at a steady pace. Of course since gravity sucks you in, you would get to that massive object much faster than you had expected, based upon how long it took you to get to that same spot when it was empty. Therefore (absent a proper understanding of higher physics) you would have to conclude that somehow the distance between the two spots had shrunk.
So it seems like the curvature would produce results suggesting the distance between the points really had shrunk, even though maybe at first glance no "space" had evaporated?