From "Abelian CAtegories and its application to Rings and Modules " by Popescu N, par. 3.5 pag 88:
Popescu using the names "small" and "finitely presented" for the yours "sumcompact" and "compact" respectively.
Popescu call a object $X$ (of a Grothendick abelian category $\mathcal{C}$)
of "finite type" if for any direct union of subobjets $Y=\cup_{i\in I}Y_i$ the natural morphism $Colim_{i\in I} \mathcal{C}(X, Y_i)\to \mathcal{C}(X, Y))$ is a isomorphism, this is equivalent to:
for any directed union of subobjets $X=\cup_{i\in I}X_i$ there is a $i_o\in I$ such that $X=X_{i_0}$.
In a category of modules finitely presented is equivalent to the usal definition (there is a exact $0\to A\to X\to C\to 0$ with $A,\ B$ finitely generated), and finite type is equivalent to finitely generated.
From 5.4 of Popescu book a finitely generated module is small (sumcompact). And of course exist finitely generated modules that aren't finitely presented. then we have the implications:
finitely presented $\Rightarrow$ finitely generated $\Rightarrow$ small (suncompact) and finitely .generated$\not\Rightarrow$ finitely presented
Then cannot have that small (sumcompact)$\Rightarrow$ finitely presented.