As you pointed out, the definitions in the literature are various, so it is difficult to have unambiguous definitions.
Usually, the book of Russ (“The Image Processing Handbook”) is a good start and contains usefull references.
Some definitions that appear rather consensual:
- “Sphericity” is often defined as a ratio of V^2/S^3 up to a constant, similarily to circularity in 2D
- “Compactness” or “Convexity” is usually ratio of volume divided by volume of convex hull
- “Elongation”, “Flatness” are usually based on ratio of lengths of the equivalent ellipsoid obtained after applying principal component analysis.
The best is to recall the definition of the parameters used before interpreting them.