In this paper, we use the notion of a differentially coprime kernel representation to parametrize the set of all nonlinear plants stabilized by a given nonlinear controller using a so-called Youla parameter and to unify understanding of some stability concepts for nonlinear systems. By utilizing the differential kernel representation concept, we are able to convert a closed-loop identification problem into one of open-loop identification. The main advantage of our approach using kernel representations over fractional descriptions is that we address a larger class of nonlinear systems. The idea of a differential kernel representation allows us also to clarify the relationship between three different notions of internal stability available in the literature. The results in the paper thus provide new insights to the stability of nonlinear feedback systems.