How do we expect distance measurements to compare inside and outside the event horizon of a black hole?

When people talk about the "diameter" they're just talking about twice the radius in some coordinate system like Schwarzschild coordinates--I talked about the physical meaning of the radial coordinate in these coordinates in this answer. The only coordinate-independent measure of "distance" is relativity is proper distance along a spacelike path (see my answer here), so you could choose some spacelike path from a point on the observer's worldline to some point in spacetime that lies on the event horizon. But you'd have to make some decision about what point you wanted to choose, the proper distance could be made arbitrarily short by picking a point that was arbitrarily close to where your past light cone intersected the horizon (since the spacetime "length" along a lightlike path, as defined by the metric, is always zero--see the spacetime wiki article if you aren't familiar with spacetime intervals, and John Rennie's answers here and here about how the metric is used to calculate proper time along timelike paths might also be helpful).

One approach to picking a spacelike path would be pick some simultaneity convention, like slices of constant time coordinate in a Kruskal Szekeres coordinate diagram, and then look at the proper distance along a spacelike curve that was confined to a single "moment", and this would correspond to a type of ruler distance, namely the sum of measurements on a bunch of short rulers whose ends line up at points along the spacelike path (I talked about the physical meaning of proper distance in that second answer of mine I linked to above) but since simultaneity is relative to your choice of coordinate system, you'd get different distances depending on what convention you chose, corresponding to different possible series of points in spacetime that the short rulers' ends line up.