Restricting a Weil divisor to a local scheme

Apologies if the title is garbage. I couldn't think up anything more pertinent, as it's really just a notational question.

In Hartshorne, p.141, Proposition 6.11, it is shown that Weil divisors are Cartier (under some assumptions).

My question is what on earth does it mean for $\mbox{Spec } \mathcal{O}_x$ to be a local scheme; moreover, how does one induce a Weil divisor $D_x$ on it?

In my head this is supposed to be restricting $D$ to a given point; the construction then gives a local equation for $D$ at $x$.

However, I am struggling to see how the scheme $\mbox{Spec } \mathcal{O}_x$ corresponds at all to $x$: it's not even a point unless the stalk is a field.

Thanks for any help offered.

A local scheme is the affine spectrum $\mathrm{Spec}(A)$ of a local ring $A$; such a scheme always has a single closed point. For any scheme $X$, the spectrum $\mathrm{Spec}(\mathscr{O}_{X,x})$ is called the local scheme of $X$ at the point $x$. There is a canonical morphism $\mathrm{Spec}(\mathscr{O}_x) \to X$ and the map of underlying topological spaces is a homeomorphism from $\mathrm{Spec}(\mathscr{O}_x)$ to the subspace of $X$ formed by the points $y$ such that $x \in \overline{\{y\}}$. All this can be found in full detail in (EGA, I, 2.4).
Let $Z = \displaystyle\sum_{x \in X^{(1)}} n_x . \overline{\{x\}}$ be a Weil divisor on $X$, where $X^{(1)}$ is the set of 1-codimensional points of $X$. For any point $y \in Y$ one defines the Weil divisor
$$Z_y = \sum_{x \in X^{(1)} \cap T_y} n_x . (\overline{\{x\}} \cap T_y)$$
on $T_y = \mathrm{Spec}(\mathscr{O}_{X,y})$. $Z$ being locally principal is equivalent to $Z_y$ being principal.