The reason for this mess is that "moment" is a generic term. For a function f(x) with domain (a,b), all the integrals (READ BOTTOM FORMULA) (PDF version) are called moments of f. The moments of mechanics are multidimensional generalizations of these.

• The center-of-mass numerators are first moments (because they involve the coordinates to the first power).
• The moments of inertia are second moments.