[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.