In a previous paper, we showed that several standard definitions of stability of nonlinear discretizations are so strong that they classify as unstable a number of useful discretizations. Then a weaker definition was introduced which, however, was powerful enough to imply, together with consistency, the existence and convergence of the discrete solutions. In this paper we prove that, for smooth discretizations, stability in the new sense is equivalent to stability of its linearization around the theoretical solution. This fact does not imply that schemes with stable linearizations are automatically useful, due to the appearance of so-called stability thresholds. The abstract ideas introduced are applied to a concrete finite-element example, with a view to assessing the advantages of the new approach.