Let $S$ be some scheme. What are the objects $\operatorname{Ob} (Sch_S)$ of the Category of Schemes over $S$, $Sch_S$?
In Vakil, we have (6.3.7) that "the objects [of $Sch_S$] are [scheme] morphisms of the form $X\rightarrow S$ [called structure maps]." Hartshorne bears this out: "a scheme $X$ over $S$ is a scheme $X$, together with a morphism $X\rightarrow S$."
Does this imply that for any one scheme $X$ there could a priori be two different objects of the category, $X\rightarrow_\phi S$ and $X\rightarrow_\psi S$? This seems odd, given that morphisms in $Sch_S$ must by definition commute with structure maps, and so these objects are not isomorphic; if they were then would have to commute: $$\require{AMScd} \begin{CD} X@>>{\simeq}> X\\ @VV{\phi}V @VV{\psi}V \\ S @>>{\simeq}> S \end{CD}$$
Is this truly the definition, or am I missing some subtlety? If this is the correct definition, how does one interpret an expression such as $\operatorname{Hom}_S(S,X)$, where $\operatorname{Hom}_S$ denotes morphisms of $S$-schemes? It seems to depend on the structure morphisms, in which case expressions (for an alg closed field $k$) like $\{x\in X \text{ closed}\}=\operatorname{Hom}_k(\operatorname{Spec} k,X)$ confuse me, as showing
$\{x\in X \text{ closed}\}\subset \operatorname{Hom}_k(\operatorname{Spec} k,X)$ seems to require that $\operatorname{Spec} k\simeq_\phi \operatorname{Spec} k$, $\phi$ the identity, is the implied structure morphism for $\operatorname{Spec} k$, whereas
$\{x\in X \text{ closed}\}\supset \operatorname{Hom}_k(\operatorname{Spec} k,X)$ seems to require that the structure morphism of $\operatorname{Spec} k$ is chosen based on the implied structure map of $X$: so that each $x$ commutes with the structure morphism of $X$ in the diagram of the canonical map $i_x: \operatorname{Spec} k=\operatorname{Spec} \kappa(x)\rightarrow X$ over $\operatorname{Spec} k$ (e.g. from this post).
Yes, the definition is correct. When you write something like $\operatorname{Hom}_S(A,B)$ then $A$ and $B$ are assumed to have fixed, chosen morphisms to $S$ that make them into $S$-schemes. In other words, in the notation $\operatorname{Hom}_S(A,B)$, "$A$" and "$B$" are abbreviations for certain morphisms $A\to S$ and $B\to S$ which are actually the objects you are considering morphisms between (in the category of schemes over $S$). This is no different from how (for instance) a topological space is actually an ordered pair $(X,T)$ where $T$ is a topology on $X$, but people almost always talk about just "$X$" as the topological space, leaving $T$ as implicit. In the same way, you talk about $A$ and $B$ being schemes over $S$, without explicitly mentioning the chosen morphism to $S$.
In the case of a statement about $\operatorname{Hom}_k(\operatorname{Spec} k,X)$, $\operatorname{Spec} k$ is being considered as a $k$-scheme via the identity map $\operatorname{Spec} k\to\operatorname{Spec} k$. Of course this is not the only possible $k$-scheme structure on $\operatorname{Spec} k$ but it is the "default" one that is assumed unless mentioned otherwise.