Natural higher genus analogues of convex surfaces are usually considered to be surfaces which satisfy the "two piece property" or are "tight".
A closed surface in Euclidean space is said to have the two piece property or be tight if no plane cuts it into more than two pieces. So for instance tori of revolution are tight as are all convex surfaces, and the polyhedral torus referenced in the question. An equivalent formulation is that the "total positive curvature", or the integral of the Gauss curvature where it is positive, must be 4π, which is its lowest possible value. This implies that all points of positive curvature must lie on the boundary of the convex hull of the surface.
The study of these surfaces originates with Alexandrov, who showed that tight analytic surfaces are isometrically rigid. Later significant contributions were made by Chern and Lashoff (who studied total curvature), Nirenberg (Who tried to extend Alexandrov's rigidity theorem to smooth surfaces), Banchoff (who coined the term "two piece property"), Kuiper, and others.
A very good reference is the survey book Tight and Taut Subamnifolds, edited by Cecil and Chern.