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?
Edit. Thanks to the answer by Leonid Positselski, now I know that the term I need is "fp-injectve", whereas "pure injective" probably means something else.