Just checking here. It is true isn't it, that the sheaf associated to a locally free presheaf of $O_X$ modules (I suppose there is an example that is not a sheaf??) over a scheme $X$ is a locally free sheaf of $O_X$ modules? If the associated sheaf is coherent then this just textbook (e.g. Hartshorne Exercise II.5.7b), but otherwise I still would suppose this is true simply by definition of locally free presheaf (i.e. free when restricted to an open set)
Sheaf associated to a locally free presheaf of modules
algebraic-geometrymodulessheaf-theory
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As Jesko says, already for line bundles there are problems. The easiest example is probably $\mathscr{O}_{\mathbf{P}^n_k}(1)$ and its dual $\mathscr O(-1)$. The first has lots of nonzero global sections, but the second has none; they can't even be non-canonically isomorphic.
One interesting thing to think about, which I saw pointed out in a book review by Kollár, is that in differential geometry one can choose a metric on any vector bundle and use that to identify $E \simeq E^\vee$. So this is a good example of how algebraic geometry differs.
Vector bundles are nice to think about, but they have problems: it is not true that the kernel and cokernel of a map of vector bundles is necessarily a vector bundle. Consider for example the ideal sheaf of the origin inside $\Bbb A^1_k$: this is a vector bundle (it's the sheaf associated to the free module $xk[x]$), and it injects in to another vector bundle $\mathcal{O}_{\Bbb A^1_k}$ (the sheaf associated to the free module $k[x]$), but the cokernel is the structure sheaf of the origin (the sheaf associated to the non-free $k[x]$-module $k$).
We would like to be in a situation where we work in an abelian category: in particular, we want to be able to take kernels and cokernels and still have them be in our category. Quasi-coherent sheaves provide one such category where we can do this, and in some sense it's the smallest possible one (the precise sense is that it's the smallest cocomplete abelian category containing vector bundles aka locally free sheaves).
This links in nicely with the "definition using exact sequences" you mention in part 2. To be precise, this definition is that locally, every quasicoherent sheaf $\mathcal{F}$ can be represented as the cokernel of a morphism of free sheaves: for every point $x\in X$ there is an open neighborhood $U\subset X$ with an exact sequence $$\mathcal{O}_X|_U^{\oplus I} \to \mathcal{O}_X|_U^{\oplus J}\to \mathcal{F}|_U\to 0$$ for some sets $I,J$.
Most reasonable notions in algebraic geometry are in some sense "local" - this means that if we want to verify that some property holds, we ought to be able to check that it holds in a neighborhood of every point. This definition of a quasi-coherent sheaf provides us the correct way to do that, and this definition is equivalent to the one above (any sheaf in the smallest cocomplete abelian category containing locally free sheaves fulfills the above definition as locally a cokernel of free sheaves and vice-versa). For some more involved discussion, you may wish to consult Vakil's FOAG, section 13.1.9, beginning on page 374, as well as this MO question, and/or this MSE question.
Best Answer
Suppose $\mathcal{F}$ is a presheaf of $\mathcal{O}_X-$modules that is locally free. This means that there exists an open cover $\{U_\alpha\}$ such that there exists $\varphi_\alpha:\mathcal{F}|_{U_\alpha}\xrightarrow{\sim}\bigoplus_{I}\mathcal{O}_{U_\alpha}$ for each $\alpha$. Such a map is an isomorphism on stalks. We know that $\mathcal{F}|_{U_\alpha}$ can be regarded as a presheaf on $U_\alpha$, and then $\mathcal{F}|_{U_\alpha}^+\cong \mathcal{F}^+|_{U_\alpha}$. By the universal property of the sheafification, the isomorphism $\varphi_\alpha:\mathcal{F}|_{U_\alpha}\to \bigoplus_I \mathcal{O}_{U_\alpha}$ induces a map $\mathcal{F}^+|_{U_\alpha}\to \bigoplus_I \mathcal{O}_{U_\alpha}.$ This map is an isomorphism: it can be checked on stalks but we know that the stalk maps were isomorphisms to begin with since $\varphi_\alpha$ was an isomorphism. So, $\mathcal{F}^+|_{U_\alpha}\cong \bigoplus_I\mathcal{O}_{U_\alpha}$.
Actually, this wasn't strictly necessary. It suffices to note that since $\mathcal{F}|_{U_\alpha}$ is isomorphic to a sheaf, it is itself a sheaf on $U_\alpha$. Hence, $\mathcal{F}^+|_{U_\alpha}\cong \mathcal{F}|_{U_\alpha}\cong \bigoplus_I \mathcal{O}_{U_\alpha}.$
By the way, the tensor product of sheaves needs to be sheafified to get a sheaf in general, so I think you can construct an example of a locally free presheaf that is not a sheaf by tensoring a pair of locally free sheaves.