I know there have been questions about this but I'm afraid I am still not really getting it. It's kind of embarrassing that something is not clicking at this point because I have asked some related questions before. I want to understand exactly what the process is for defining a Weil divisor from an invertible sheaf along with a choice of "rational section".
Throughout, let $X$ be a noetherian, integral, normal scheme.
My first confusion is precisely what is meant by rational section. Most sources just define it as "a section of $\mathcal{L}$ defined over a dense open subset". This seems to just be the definition of an element of the stalk of $\mathcal{L}$ at the generic point. Is there a reason they don't just define it this way, or is it somehow different?
In the case that $\mathcal{L}$ is just the structure sheaf, then a rational section is just a non-zero element of the quotient field of any affine open subset, call it $f \in K^{*}(X)$. Then if $Y$ is a prime Weil divisor with generic point $\eta$, then the local ring $\mathcal{O}_{\eta} \subseteq K^{*}(X)$ is a DVR and we obtain a valuation corresponding to it. I am having trouble reducing the invertible sheaf case to this.
Let $(\mathcal{L}, s)$ be an invertible sheaf and rational section (which I am assuming is just an element of the stalk of $\mathcal{L}$ at the generic point of $X$). Now if $Y$ is again a prime Weil divisor with generic point $\eta$, we can choose a trivialising open neighbourhood $U \ni \eta$ with trivialisation $$ \psi: \mathcal{L}|_{U} \longrightarrow \mathcal{O}_{U}. $$ But it is not clear where to go from here. I want to somehow transport the element of the stalk of $\mathcal{L}$ to an element of the stalk of $\mathcal{O}_{U}$. So do I just take the colimit of this trivliaisation over open subsets inside $U$? Can someone walk me through this process in detail, because I find a lot of resources just seem to handwave and say something like "just transport the rational section" and it isn't clear to me what that means in rigorous terms.