Let Ext(C,A) be the group of extensions of A by C and Pext(C,A) the subgroup of Ext(C,A) containing pure extensions. Is there a proper subgroup of Ext(C,A) containing Pext(C,A)?
are you looking for a categorical definition that is always between the two (which likely does not exist), or some examples (which likely do exist)? a short investigation leads to a book by laszlo fuchs that may answer your question (since there are apparently many cases in which pext(C, A) = 0)
See the linked to PDF for reference.
Proposition 5.6 gives you the information to build an example that it is indeed possible.
From Proposition 5.6.1, you get a way of writing $Pext$ as an intersection of kernels of maps $Ext$ and another group, so if you can create an example where this intersection is non-trivial, then you've got your answer.
On the other hand, maybe you're more familiar with inverse limits. Then in Proposition 5.6.3 there is a short exact sequence (See 5.7). If the inverse limit on the right hand side is non-zero, then you have an example where $Pext$ can be identified with a proper subgroup of $Ext.$
http://nyjm.albany.edu/m/2003/1p.pdf
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