In Stanford Encyclopedia, concerning the semantics of second order logic, Herbert B. Enderton wrote
... an assignment $s$ of objects to the free variables in $\phi$. ... For a $k$-place predicate variable $P$,
$M \models \forall P\; \phi[s]\;$ iff$\;$ for every $k$-ary relation $Q$ on $A$, we have $M \models \phi[s']$
where $s'$ differs from $s$ only in assigning the relation $Q$ to the predicate variable $P$.
I think I understand informally what this means, but I am stuck when I try to understand how it works formally. Does the original assignment $s$ also assigns a value to the predicate variable $P$? Naively, I would say no, because otherwise $\forall P$ is uninteresting (or else I misunderstand how assignments work), but the context seems to say the opposite.
Also, why do we need a second use of $\models$. Couldn't we say
$$M \models \forall P\; \phi[s]\; \iff \forall Q\; \phi^M[s]$$
where $\phi^M$ is the result of substituting the symbols in $\phi$ by their interpretation fixed in $M$, $P$ by $Q$, and the range of $Q$ is also fixed in $M$ (the possible ranges depend on the semantic) and $s$ assigns a value to all free variables in $\phi$, except $Q$, which is taken care by $\forall Q$.
I use the assumption that the formal language is very close to the language that is used to define the models, both have quantifiers, etc. So, I am trying to leverage on this. May be it is too naive. What am I missing?
I am trying to understand how the formalism works. Also, again, why we need a second use of $\models$. If it was third order logic, would we need to use it three times? These two questions are most likely related. As usual, I must be missing some thing basic.