Suppose $f：X \to Y $ is a morphism. I saw two definitions of scheme theoretic image.
The first one requires $f$ to be quasi-compact and quasi-separated, or quasi-compact, which ensures the kernel $J=O_Y\to f_*(O_X)$ be quasi-coherent. Thus $J$ defines a closed subscheme of Y.
The second one requires nothing, it define one by the quasi-coherent sheaf $\sum J_W$ where $J$ is the defining ideal of $W$, and $W$ runs through all the the closed subschemes $W$ which $f$ factors through.
What is there difference?
As Martin noted, any definition of scheme-theoretic image will be compatible with the second definition you give.
The point is that in the absence of some (mild!) assumption on $f$, it is impossible to prove anything about scheme-theoretic image; in paticular, it is impossible to prove (since it won't be true in general) that it is local on the base (i.e.\ that for an open subset $U$ of $Y$, the sch.th.im. intersected with $U$ coincides with the sch.th.im. of $f_{|f^{-1}(U)} : f^{-1}(U) \to U$.
So people normally only speak of sch.th.im. when $f$ satisfies a hypothesis (such as quasi-compactness) which allows this property to be verified.