You are asking for the completeness theorem of first-order logic, proved by Kurt Gödel in 1929.
There are various ways to state the completeness theorem, and among them are the following two assertions:
Whenever a statement $\varphi$ is true in every model of a theory $T$, then it is derivable from $T$.
Whenever a theory $T$ is consistent, then it has a model.
These assertions are easily seen to be equivalent, by the following argument. If the first holds, and a theory $T$ has no model, then false holds (vacuously) in every model of $T$, and so $T$ derives a contradiction; so the second holds. If the second holds, and $\varphi$ holds in every model of $T$, then $T+\neg\varphi$ has no models and so is inconsistent by 2, so by elementary logic, $T$ derives $\varphi$; so the first statement holds.
Your confusion is a good one: when people say e.g. "$\omega$ is absolute between transitive models of $\mathsf{ZFC}$," what they really mean is "the formula $\varphi_\omega$ is absolute between transitive models of $\mathsf{ZFC}$" where $\varphi_\omega$ is the usual formula defining $\omega$ in $V$.
Note the bolded clause, however - there are multiple ways to define $\omega$ in $V$, and not all of them are absolute! For example, suppose $\mathsf{CH}$ holds in $V$, and consider the formula $$\psi(x)\equiv (\mathsf{CH}\rightarrow \varphi_\omega(x))\wedge(\neg\mathsf{CH}\rightarrow \forall y(\neg y\in x)).$$ This still defines $\omega$ in $V$, but is clearly not absolute between transitive models of $\mathsf{ZFC}$: if $M\models\mathsf{ZFC+\neg CH}$ is transitive then $\psi^M=\emptyset$.
So actually there's a huge potential for nonsense here, and this conflation of an object with some fixed definition of it is a really annoying abuse of notation. Fortunately it's rarely an issue, since generally there's only one reasonable choice of definition and it's clear from context, but it's still very much an abuse. (And every so often it does matter - less in terms of proving results than in developing intuitions, but still.)
The same is true for (class) functions: we often conflate an object with some fixed formula defining it. And again, this depends crucially on what definition we pick.
Meanwhile I don't really understand your last paragraph: it sounds like you're asserting a difference between "$\varphi$ is true in $M$" and "$M$ thinks $\varphi$ is followed," but I don't see what that difference would be.
(Note that, following the above abuse of notation, we do have $V^M=M$: the "standard" formula defining $V$ is $x=x$, and $M\models\forall x(x=x)$. So in fact if $M\models\theta$ then $M$ thinks $\theta$ is true in $V$ - since $M$ thinks it is $V$. Maybe that's what you're getting at?)
Best Answer
I'd interpret it as a definition schema, where you could interpret "$\phi$ is absolute for $\mathbf M,$ $\mathbf N$" as an abbreviation for the sentence $$ \forall x(x\in \mathbf M\to x\in \mathbf N) \land \forall \vec x\in \mathbf M(\phi^{\mathbf M}(\vec x)\leftrightarrow \phi^{\mathbf N}(\vec x)).$$
One could ask the same question of "what definition of true" is being used when we say "$\mathbf M$ is transitive" (also a schema of sentences, but in one formula rather than three). Generally, we are working with some reference set theory (say maybe $\sf ZF$, maybe without regularity at this point in Kunen's book), and when we work with such statements, we're allowed to use those axioms.
With that said, looking through the next couple of pages, I don't think I see anything that can't be proved just with pure logic, e.g. I don't think the theorem that syntactically $\Delta_0$ formulas are absolute or Lemma 3.7 require any set theory axioms (but I could have missed something).
Speaking of Lemma 3.7, since you asked about that too: Here, $S$ is a set of sentences in the metatheory and the statement that $S\vdash \phi\leftrightarrow \psi$ is a statement in the metatheory. Let $\sf WST$ ("working set theory") be the reference theory I alluded to earlier. What this metatheorem is saying is that if $S\vdash \phi\leftrightarrow \psi$, and $\sf{WST}\vdash\varphi^{\mathbf M}$ for each $\varphi\in S$, and similarly for $\mathbf N,$ then $\sf WST$ proves the sentence "$\phi$ is absolute for $\mathbf M,$ $\mathbf N$ iff $\psi$ is."
As I remarked above, I don't think anything other than logic is used in proving lemma 3.7, but set theory axioms will almost always be used when using it, at least assuming $S$ is not the empty set. For example, we'll often need to use the corresponding axiom $\varphi$ in $\sf WST$ to prove $\varphi^{\mathbf M}$ for $\varphi\in S,$ or need to use some set theory axioms in any event.
As Kunen mentions, the main purpose of lemma 3.7 is to transfer absoluteness for syntactically $\Delta_0$ formulas to absoluteness for formulas/concepts provably equivalent to a syntactically $\Delta_0$ formula in some set theory $S$ (or more generally, $\Delta_1^S$ concepts... those provably equivalent to both a syntactially $\Sigma_1$ and a syntactically $\Pi_1$ formula in $S$). Then this allows you to establish (schematically, in $\sf WST$) that the concept is absolute for transitive models of $S.$
So when Kunen later proves a bunch of things are absolute for transitive models of $\sf ZF-P$, or what-have-you, we're really concerned about scrutinizing $S$ rather than $\sf WST.$ Though we'll have $S\subseteq \sf WST$ here since we have $\mathbf N = \mathbf V.$ For instance one of the more important and not-altogether-trivial results is that being an ordinal is absolute for transitive models. This requires foundation since we need the equivalence of being an ordinal to being transitive and linearly ordered by $\in$ (rather than well-ordered by $\in$, which is the 'real' definition that works in weaker set theories).