There are at least two ways to define a (quasi)coherent sheaf on the scheme $(X,\mathcal{O}_X)$, namely the one given in Hartshorne's "Algebraic geometry" (I guess you know it: it is given in terms of "abstract" modules) and the second close to Wikipedia's one: $\mathcal{O}_X$-module $\mathcal{F}$ is quasicoherent if and only if each point has a neighbourhood $U$ such that the sequence $$\bigoplus_{i\in I} \mathcal{O}_U\to \bigoplus_{j\in J} \mathcal{O}_U\to \mathcal{F}|_{U}\to 0$$ is exact. Moreover, $\mathcal{F}$ is coherent on noetherian $X$ if $J$ can be chosen finite.
How can I prove these definitions are equivalent? I'm sorry if the question is childish: I am a beginner in schemes and such kind of geometry.
The other definition is that there's an affine cover on each set $U=\text{Spec } A$ of which $\mathcal F$ is the cokernel of a map of sheaves of the form $\widetilde{\mathcal M}\to \widetilde{\mathcal N}$. The cokernel of a sheaf map is not generally a sheaf, but $\widetilde{\bullet}$ is an equivalence of categories from $A$-modules to quasicoherent $\mathcal{O}_A$-modules, so the cokernel must be $\widetilde{M/N}$. But we can write $M/N$ as the cokernel of a map of free $A$-modules (that's what a presentation is,) so we can $\widetilde{M/N}$ as the cokernel of a map of free $\mathcal{O}_A$-modules.