Distributionally robust optimization hedges a decision against an ambiguity set of candidate distributions. In a cold-start setting the target distribution is unobserved and no sample from it is available, so only related source distributions provide evidence for constructing that set. We propose an inclusive-exclusive Wasserstein ball model that jointly learns a center and radius from inclusive sources believed compatible with the target and exclusive sources believed incompatible with it. Soft penalties balance enclosure and exclusion against ball size and margin. We identify parameter conditions for unboundedness and radius collapse, and show that small regularization permits an exact relaxation of the radius constraint. In this regime the Lagrangian dual involves a signed Wasserstein barycenter, and, assuming an exact signed-barycenter oracle, we propose a cutting-plane learning algorithm that constructs the ball and certifies bounds on the dual optimum and on the suboptimality of candidate balls. For quadratic transport under small regularization, we establish exact convex reformulations for univariate laws and for translations and scalings of a common base law. These yield finite-dimensional convex quadratic programs for univariate empirical sources and common-base families, with explicit recovery of a globally optimal ball. The learned ball then serves as the ambiguity set of the downstream robust out-of-distribution stochastic decision.