Hmm. I only see 'intrinsic curvature' used when there is a potential confusion about whether it's intrinsic or extrinsic (otherwise it's just 'curvature'). And if that is the case, so that there is an embedding in a higher dimensional space, then intrinsic/extrinsic curvature are excellent terms (imo). They were some of the first math terms I came across where I really appreciated the selection of terminology.