I am interested in the following assumption on left $R$-modules: for a module $I$ and all injective homomorphisms $A\to B$ of finitely generated (or possibly finitely presented) modules I want the homomorphism $Hom_R(B,I)\to Hom_R(A,I)$ to be surjective. Is this condition strictly weaker than the injectivity of $I$; how can one construct examples of this sort?

What is the relation of my condition to pure injectivity of $R$-modules? I do not understand its relation to the "standard" definition of the latter notion; also, what is the relation of the "standard" definition to Terminology 11.1 in the paper "Relative Homological Algebra and Purity in Triangulated Categories" of Beligiannis?