Ring homomorphisms which yield same endomorphisms on modules

A map of rings is an epi iff the corresponding restriction-of-scalars functor is full.

I have no idea of what might characterize maps which are full only on endomorphism rings... But in any case it is a slightly unnatural question, no?

A relevant (perhaps more natural) question to mine is when all homomorphisms are the same, not just endomorphisms. That is, can we find a (nice) condition on $\varphi$ such that for every pair of $S$-modules $M$ and $N$ the inclusion $$\text{Hom}_S(M,N) \subseteq \text{Hom}_R(M,N)$$ becomes an equality?

Yes, this happens iff $R \to S$ is an epimorphism in the category of rings. Details and much more stuff on epimorphisms in the category of commutative rings can be found in the corresponding Séminaire Samuel. It is already quite interesting to characterize epimorphisms when $R=\mathbb{Z}$ (or more generally a Dedekind domain), see here.