This is not pure-injectivity.  The relevant concept is that of an fp-injective ("finitely presented-injective") module, otherwise known as an "absolutely pure" module.  A left $R$-module $J$ is said to be fp-injective if for any finitely presented left $R$-module $M$ one has $Ext^1_R(M,J)=0$.  

The notion of an fp-injective left $R$-module is particularly well-behaved when the ring $R$ is left coherent.  Over a left Noetherian ring $R$, fp-injectivity is equivalent to injectivity.

References:

1. B.Stenström, "Coherent rings and $FP$-injective modules", Journ. London Math. Soc. vol.2, 1970, https://doi.org/10.1112/jlms/s2-2.2.323

2. my paper L.Positelski "Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality", Selecta Math. vol.23, 2017, https://arxiv.org/abs/1504.00700