[danbruc]

The concept of "curvature of space" is obvious nonsense because curvature is measured with respect to space.

This is not really true. If you take a flat two dimensional space, a sheet of paper, this space has no intrinsic curvature. You surely now imagine this sheet of paper laying flat on a table in front of you embedded in our usual three dimensional space. Now pick it up and role it to form a cylinder. This gives extrinsic curvature to the space, curvature in the space the sheet is embedded in. But the sheet has still no intrinsic curvature, if you were a two dimensional creature living on the sheet you can not tell whether it is laying flat on the table or is rolled up to a cylinder, at least ignoring the fact that the cylinder connects two opposite edges of the space.

The surface of the earth on the other hand has intrinsic curvature, angles of triangles don't add up to 180 degrees for example. And this intrinsic curvature is a feature of the space not of the embedding into a higher dimensional space. It is not easy to visualize if possible at all, but spaces can have intrinsic curvature independent or even without an embedding, our three dimensional space can be curved without being embedded in a higher dimensional space and the same of course holds for space-time.

[mrob]

"Extrinsic curvature" is normally called just "curvature". "Intrinsic curvature" isn't normally called anything because it has minimal relevance to everyday life. If you talk about "curvature of space" the natural assumption is that you mean extrinsic curvature.

[danbruc]

But in general relativity extrinsic curvature is the irrelevant thing, it's all about intrinsic curvature. We don't think of space-time being embedded in a higher dimensional space but we still talk about curvature of space-time, its intrinsic curvature.

These demonstrations with balls rolling on a stretchable rubber sheet to visualize gravity are really misleading in this regard because they use a good deal of extrinsic curvature to make things work but in reality mass doesn't deform space-time into a fifth dimension or at least it doesn't necessarily do so.