Let $X$ defined over a field $k$ be a smooth proper curve, and suppose I have a vector bundle in the sense of alebraic geometry $E \to X$. Then $E$ corresponds to a locally free $\mathcal{O}_X$-module of rank $n$. Let $\eta$ be the generic point of $X$ and let $F_\eta=k(X)$ be the function field of $X$. Then in principal, one may always obtain a natural trivialization over the generic fiber $E_{\eta} \to \operatorname{Spec}(F_\eta)$, because the pullback of an $\mathcal{O}_X$-module to $\operatorname{Spec}(F_{\eta})$ is just $E \otimes _{\mathcal{O}_X}F_{\eta} \cong (F_{\eta})^n$. This is a trivial bundle over $\operatorname{Spec}(F_{\eta})$ and so is $\cong F_{\eta}[x_1,\ldots,x_n]$.
Here is where my confusion starts: For any schemes $S,T$ of finite presentation over $X$, and any morphism of generic fibers $f_{\eta}: S_{F_{\eta}} \to T_{F_{\eta}}$ then $f$ spreads out to a morphism over some open dense $U$ in $X$: $f_{U}:S_{U} \to T_{U}$. Usually the way one sees this is one writes down the polynomial map over $F_{\eta}$ and then my avoiding the poles one obtains the dense open $U$. In this case of the vector bundle, I just have the induced structure morphism $F_{\eta} \to F_{\eta}[x_1,\ldots,x_n]$ there is no polynomials to argue with and it feels wrong to conclude that I can know from this which open $U$ my morphism spreads out to.
Now you might say that this data is hidden in the choice of isomorphism $E \otimes _{\mathcal{O}_X}F_{\eta} \cong (F_{\eta})^n$, indeed such an iso is given by a matrix of rational functions and so an element of $GL_n(F_{\eta})$ and this can define the $U$. Again here I'm confused because although the line/vect bundle shouldn't depend on isomorphism class, and so changing the trivialization over the generic fiber is like multiplying by a principal divisor in the case $n=1$, or $GL_n(F)$ element for rank $n$, thus we can multiply by a suitable element and clear all the poles. I should be free to apriori assume by reasoning up to isomorphism that the isomorphism over the generic fiber is "defined over $\mathcal{O}_X$. Why is this reasoning wrong?