I am looking into the definition of the $S$-matrix, and have found two different cases. Firstly I have seen it derived that (see here, link to Google books p110): $$ S=U_S(\infty,-\infty)$$ But more commonly that (see here, link to Google books p169): $$S=U_I(\infty,-\infty)$$ (Where subscript $S$ represents the Schrödinger picture and $I$ the interaction). Are these two definitions identical? If so can this be shown and if not why two different definitions? Either way the latter seems to be preferred, why?
Further Information/My guess:
This may come down to how we define $S$ from what I can tell it is define such that if your particle is in the Schrödinger state $|\psi_S(t)\rangle$ and you want to find the amplitude that at $t=\infty$ it is in the state $|\phi_S(t)\rangle$ then $S$ is defined such that: $$\langle\psi_S(\infty)|S|\psi_S(-\infty)\rangle=\langle\psi_S(\infty)|\psi_S(\infty)\rangle$$ which would indicate my former rather then my latter definition. Thus different authors may be defining it in different ways?