Given a scheme $(X,\mathcal O_X)$ and a sheaf $\mathcal F$ of $O_X$-Modules, the following are equivalent:
a) There exists a covering $\mathcal U=(U_i)$ of $X$ by open subsets $U_i\subset X$ and $\mathcal O_{U_i}$-isomorphisms $\mathcal F|U_i \simeq \tilde M_i$ for some family of $\mathcal O(U_i)$-modules $M_i$.
b) For every affine open subset $U\subset X$ there exists an $\mathcal O(U)$-module $M$ ( namely $M=\mathcal F (U)$) and an $\mathcal O_{U}$-isomorphism $\mathcal F|U \simeq \tilde M.$
This equivalence is a theorem, proved for example in Mumford's Red Book, at the very beginning of Chapter III, in §1 (along with other equivalent characterizations). This has nothing to do with noetherian hypotheses.
The sheaves satisfying these equivalent conditions are called quasi-coherent
and this definition is unambiguous and undisputed.
And now on to coherent sheaves.
Recall that a a sheaf $\mathcal F$ of $O_X$-Modules is said to be finitely generated if for every $x\in X$ there exists an open neighbourhood $U$ of $x$ and a surjective sheaf homomorphism $\mathcal O_{U}^r \to \mathcal F|U \to 0$ for some integer $r$. The sheaf $\mathcal F$ is then said to be coherent if it is finitely generated and if for every open subset $V\subset X$ and every (not necessarily surjective !) morphism
$\mathcal O_{V}^N \to \mathcal F|V$, the kernel is also finitely generated . Again, no noetherian hypothesis in sight. End of story? Not at all! The problem is that coherence is very difficult to check in general and actually for some schemes, even affine ones, the structure sheaf $O_X$ is not coherent, and in that sad case the concept coherent is essentially worthless . In particular, and this one of your questions, the equivalence of categories mentioned in Corollary (5.5) is FALSE without the noetherian hypothesis.
However all troubles evaporate if you assume that $X$ is locally noetherian. You then have the wonderful equivalence (implying of course that the structure sheaf $O_X$ is coherent)
$$\mathcal F \;\text {coherent} \stackrel {X \text {loc.noeth.}}{\iff} \mathcal F \; \text {finitely generated and quasi-coherent }$$
Edit I have tried to evade the issue, but since Li explicitly asks: Yes, Hartshorne's definition is incorrect. Here is what I mean.
The notion of coherent sheaf was introduced by Henri Cartan in the theory of holomorphic functions of several varables around 1944. In 1946 Oka proved that $\mathcal O_{\mathbb C^n}$ is coherent and this is a very difficult theorem, not following at all from Cartan's definition, the one I reproduced above.
In 1955, as is well known, Serre introduced coherent sheaves into Algebraic Geometry in his famous article Faisceaux Algébriques Cohérents and used the exact same definition as Cartan, as acknowledged in his Introduction.
Coherent sheaves were then defined in EGA for schemes and ringed spaces, always with Cartan's definition above. Ditto for the generalized analytic spaces (with nilpotents) introduced by Grauert (influenced by Grothendieck) around 1960. And that definition is also the one used in De Jong and collaborators's recent monumental online Stacks Project.
So the definition I reproduced above is the one adopted by the founders and in the foundational documents. To change it would be, in my opinion, very misleading and might for example induce one to believe that very profound theorems are trivial. Or worse, induce mistakes by inappropriately applying results from texts using the standard definition of "coherent sheaf".
Incidentally, Mumford very elegantly solves the definition problem: he only defines "coherent" in the noetherian case since he only only uses the notion in that case!
If $X=Spec(A)$, we have $X_ \text {red}=Spec(A_\text {red})$, where $A_{\text {red}}=A/\text{Nil(A)}$ so that:
$\Gamma(X,\mathcal O_X)=A$
but
$\Gamma(X_{\text {red}},\mathcal O_{X_ \text {red}})=A_{\text {red}}=A/\text{Nil(A)}$
No contradiction with Hartshorne's 2.2.2.(c)!
Edit: some details
Here are some statements which might help shed light on this subtle question.
a) Given a scheme $X$ we associate to it the quasi-coherent sheaf of ideal $\mathcal N\subset \mathcal O_X$ defined for an arbitrary open subset $U\subset X$ by $$\mathcal N(U)=\{f\in \mathcal O_X(U)\mid \forall x\in \mathcal O_{X,x },\; f_x \in \text {Nil}(\mathcal O_{X,x }) \}$$ b) The scheme $X_{\text {red}}$ has structure sheaf $\mathcal O_{X_{\text {red}}}=\mathcal O_X/\mathcal N$
c) For any affine subset $U=\text {Spec} (A)\subset X$, we have $ \text {Nil}(\Gamma(U,\mathcal O_X))=\mathcal N(U)=\text {Nil}(A)$
d) For any affine subset $U=\text {Spec} (A)\subset X$, we have $\mathcal O_{X_{\text {red}}}(U)=A_{\text {red}}=A/{\text {Nil}} (A)$
e) For a general open subset $U\subset X$ , we have $ \text {Nil}(\Gamma(U,\mathcal O_X))\subset \mathcal N(U)$
but the inclusion may be strict for non-affine $U$:
Let $X_m=\text {Spec}(\mathbb C[T]/T^m)=\text {Spec}(\mathbb C[\epsilon _m])$ and $X=\bigsqcup X_m$ (a non-affine scheme).
Then $\Gamma(X,\mathcal O_X)=\prod \Gamma(X_m,\mathcal O_{X_m})=\prod \mathbb C[\epsilon _m]$ and for $\epsilon=(\epsilon_1,\epsilon_2,\cdots)$ we have $\epsilon \notin \text {Nil}(\Gamma(X,\mathcal O_X))$ although $\epsilon \in \mathcal N(X)$.
Best Answer
Pick a finite number of generators of $M_i$ as an $A_i$ module. Since the restrictions of global sections generate $M_i$, we can write each generator of $M_i$ as a finite $A_i$-linear sum of the global generators (this works because we can only consider finite sums!). Thus we get a finite list of global generators which suffice to generate $M_i$. Now take the union of the lists for each $M_i$ across all elements of your finite cover, and you get a finite list of global generators for your global sheaf.