In this question I want to ask for a better proof than the one I am about to give for the statement with finitely presented, and inquiry if the statement is also true for the notion of finitely related (from my proof it's not clear at all that it should be, since I used explicitly finitely generatedness).
Let $(f_i)$ be elements in $A$ such that $(f_i)=A$ and assume that for an $A$-module $M$ we have the following presentations for all $i$: $$A_{f_i}^{\mu_i}\xrightarrow{\psi_i} A_{f_i}^{\eta_i}\xrightarrow{\phi_i}M_{f_i}\to0.$$ To show that $M$ is a finitely presented $A$-module.
Proof:
It's not hard to get a finite generation of M of the form $\oplus_{i=1}^n A^{\eta_i}\xrightarrow{\Phi=\Sigma f_i^J\phi_i}M$, for a suitably large $J$ (in particular $J$ has to be large enough that $f_i^J\phi_i(a)\in M$ for $a\in A^{\eta_i}$. I leave the other details to the reader).
Now, if the kernel of each one of the $\phi_i$'s is $A_{f_i}^{\mu_i}\xrightarrow{\psi_i}$, I really want to say that the kernel of $f_i^J\phi_i|_{A^{\eta_i}}$ is $A^{\mu_i}\xrightarrow{f_i^J\psi_i}$ (is it true? I'm not entirely convinced).
On the other hand, the kernel of $\oplus_{i=1}^n A_{f_i}^{\eta_i}\xrightarrow{\tilde\Phi=\Sigma f_i^J\phi_i}M_{f_1,...f_n}$ should be $\left(\bigoplus_{i=1}^n A_{f_i}^{\mu_i}\right)\oplus A_{f_1,...,f_n}^\Gamma$, where $A_{f_1,...,f_n}^\Gamma$ is generated by the finitely many relations $\left(f_i^J\phi_i\right)^{-1}(g_k)\sim \left(f_j^J\phi_j\right)^{-1}(g_k)$ where $(g_i)_{i=1}^n$ is a generating set for $M$.
Now if my argument holds, then it should not be difficult to get a finitely generated kernel for $\Phi$ from that of $\tilde\Phi$. But is there a cleaner proof? And can you get a proof for finitely related as well? (in the last step I explicitly used a finite generating set for $M$).
I haven't read your proof, sorry. Here is a quick proof. W.l.o.g. $I$ is finite.
Lemma 1: If $M \to N$ is an epi/mono/isomorphism when localized at each $f_i$, then the same is true for $M \to N$.
Proof: If $K_{f_i}=0$ for all $i$, then an elementary argument yields $K=0$. Now apply this to the cokernel or/and kernel of $M \to N$.
Lemma 2: If $M_{f_i}$ is finitely generated for all $i \in I$, then $M$ is finitely generated.
Proof: Choose $k_i$ generators of $M_{f_i}$, w.l.o.g. consisting of elements in $M$. This gives a homomorphism $A^{k_i} \to M$ which is an epimorphism when localized at $f_i$. Then $\oplus_i A^{k_i} \to M$ is an epimorphism when localized at each $f_i$, i.e. it is an epimorphism by Lemma 1.
Lemma 3. If $M_{f_i}$ is finitely presented for all $i$, then $M$ is finitely presented.
Proof: By Lemma 2 $M$ is finitely generated. Choose an exact sequence $0 \to K \to F \to M \to 0$ with $F$ finitely generated free. By assumption $K_{f_i}$ is finitely generated for each $i$ (here I use the basic fact that finite-presentation doesn't depend on the generating set!), hence $K$ is finitely generated by Lemma 2. This means that $M$ is finitely presented.
I'm not sure if the result holds for finitely related modules (probably not).