Defining spatial operators (Equals, Disjoint, Touches, Within, Overlaps, Crosses, Intersects, Contains, Relate)?

You're looking for the "Dimensionally Extended 9 Intersection Matrix" or DE-9IM for short.

DE-9IM by FME

That FME link has great examples of the spatial operators you listed above. It breaks it down into a 3x3 true/false matrix with examples and descriptions of each predicate attribute.

The reference work I use for expectations of spatial operators is the Clementini paper ("A Small Set of Formal Topological Relationships Suitable for End-User Interaction", Eliseo Clementini, Paolino Di Felice, and Peter van Oosterom, 1993). It lays out the theory behind the operators with respect to interiors, exteriors, and dimensionality, which eliminates the guesswork of which relationships are captured by which operators between which types.

Whether all implementations honor all aspects of Clementini operators is, unfortunately, another issue.