Let $X$ be an scheme and $F$ be a quasi coherent sheaf on $X$. There are three ways to consider flatness in the category of quasi coherent sheaves Qco($X$). 1- $F$ is called flat if $F_x$ is flat for each $x$ in $X$. (This definition is more usual) 2- $F$ is flat if $F(u)$ is a flat $O(U)$-module for each open affine subset $U$ of $X$. 3-(categorical definition) In the case that $X$ is semi-separated quasi-compact scheme, Qco($X$) is a locally finitely presented category. So one can think of flat objects via purity. $F$ is called flat if for each exact sequence $$0\to M\to N\to F\to 0$$ of quasi coherent sheaves, the induced sequence $$ 0\to Hom(P,M)\to Hom(P,N) \to Hom(P,F)\to 0$$ of abelian groups is exact for each finitely presentedquasi coherent sheaf $P$.
\remark: Note that when $X$ is semi separated and quasi compact, $P$ is finitely presented if and only if it is coherent.(Murfet notes, therisingsea.org)
IS there any relation between these classes of quasi-coherent sheaves?