Given a coherent sheaf $F$ over a space $X$, I understand from https://en.wikipedia.org/wiki/Support_of_a_module that the support of the sheaf of modules is all those points of $X$ such that the stalk at $x$ is not $0$.
Similarly, I can see how an open set of $X$, say $U$, can have a notion of support as a restriction, simply those elements of $U$ that again have non-trivial stalks.
My confusion is around the definition of support that takes into account a particular section over $U$, say $s$. Both here Support of a global section closed and in Mumford's Red Book, the definition is given as
$$\text{Supp}(s) = \{ y \in U | s_y \neq 0 \} $$
What exactly does $s_y$ here mean? It can't be just, those elements of $U$ whose stalk is non-trivial, because that wouldn't "depend" on a choice of $s$. Insofar as a sections are a generalization of a "ring of functions" maybe it's like $s$ "evaluated" at $y$?
Let me try to clear things up:
There are several different definitions at play here. As you say, the support of a sheaf $\mathcal{F}$ over a space $X$ is defined as $$ \operatorname{Supp}(\mathcal{F})=\{P\in X\ |\ \mathcal{F}_P\neq 0\}. $$ This behaves nicely under restriction, namely if $U\subseteq X$ is an open set, we have $$ \operatorname{Supp}(\mathcal{F}|_U)=\operatorname{Supp}(\mathcal{F})\cap U. $$ Now there are two basic definitions for the support of a section. If $s\in\mathcal{F}(U)$, then we can define $$ \operatorname{Supp}(s)=\{P\in U\ |\ s_P\neq 0\} $$ where $s_P$ is just the image of $s$ in the stalk $\mathcal{F}_P$. This sure does depend on $s$, and I don’t see why it shouldn’t: after all, you’re just looking at one particular function of you’re space of functions and your looking at where it’s nonzero. Whereas for the support of a sheaf, you are taking into account a whole set of functions. This can be seen in the following equality: $$ \operatorname{Supp}(\mathcal{F}|_U)=\bigcup_{\substack{V\subset U \\ s\in\mathcal{F}(V)}}\operatorname{Supp}(s) $$
Note that all of the above can be defined for a general topological space $X$ and sheaf of abelian groups $\mathcal{F}$.
Now you also mentioned “evaluating” a section at a point. This usually means something slightly different than just looking at the image in a stalk: namely, if $X$ is a scheme and $\mathcal{F}$ an $\mathcal{O}_X$-module, then the value of a section $s\in\mathcal{F}(U)$ at a point $P\in U$ is defined to be $$ s(P):=s_P+\mathfrak{m}_P\mathcal{F}_P\in \mathcal{F}_P/\mathfrak{m}_P\mathcal{F}_P, $$ i.e. the image of $s$ in the quotient of $\mathcal{F}_P$ by $\mathfrak{m}_P\mathcal{F}_P$, where $\mathfrak{m}_P$ is the maximal ideal of the local ring $\mathcal{O}_P$. This is because e.g. if $X$ is a variety over an algebraically closed field $k$, we want to see the sections on $U$ as an honest function from the closed points of $U$ to $k$. If “evaluating” a section $s\in\mathcal{O}_X(U)$ would mean just looking at the image in the stalk, then the “value” of $s$ would lie in the local ring $\mathcal{O}_P$. But if we define the value as the image in the residue field $\mathcal{O}_P/\mathfrak{m}_P \mathcal{O}_P=k$, then we obtain a value in $k$.
Thus, there is unfortunately a second notion of support of a section. Namely, in the situation described above, we can also define the support of $s$, resp. to avoid confusion let me call it the non-vanishing locus, as $$ D(s)=\{P\in U\ |\ s(P)\neq 0\}. $$ This is always an open set, whereas $\operatorname{Supp}(s)$ is always closed in $U$, and we always have $D(s)\subseteq\operatorname{Supp}(s)$. It is not always true that $\overline{D(s)}=\operatorname{Supp}(s) $, but for example if $X$ is reduced and $s\in\mathcal{O}_X(U)$ then this is true. See also this post.