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*.