We consider the problem of Lipschitz classification of singularities of Real Surfaces definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic) with respect to the outer metric. The problem is closely related to the problem of classification of definable functions with respect to Lipschitz Contact equivalence. Invariants of bi-Lipschitz Contact equivalence presented in Birbrair et al. (2017) are used as building blocks for the complete invariant of bi-Lipschitz equivalence of definable surface singularities with respect to the outer metric.