Some comments about your critique regarding some "imperfections" in Marker's treatment.
The basic semantics definitions are [see page 11] :
Definition 1.1.6 Let $\phi$ be a formula with free variables from $\overline v = (v_{i_1}, \ldots, v_{i_n})$, and let $\overline a = (a_{i_1}, \ldots, a_{i_n}) \in M^m$ [where $M$ is teh universe (or domain) of the $\mathcal L$-structure $\mathcal M$]. We inductively define
$\mathcal M \vDash \phi(\overline a)$ as follows [...].
Then [page 14] :
An $\mathcal L$-theory $T$ is simply a set of $\mathcal L$-sentences.
And [page 18] :
Definition 1.2.12 Let $T$ be an $\mathcal L$-theory and $\phi$ an $\mathcal L$-sentence. We say that $\phi$ is a logical consequence of $T$ and write $T \vDash \phi$ if $\mathcal M \vDash \phi$ whenever $\mathcal M \vDash T$.
Thus, the definition at page 72 :
Definition 3.1.1 We say that a theory $T$ has quantifier elimination if for
every formula $\phi$ there is a quantifier- free formula $\psi$ such that
$T \vDash \phi \leftrightarrow \psi$.
The "problem" - as you noted - is that there is no definition of logical consequence for formulae in general.
If we consider Theorem 3.1.4 :
$T \vDash \forall \overline v(\phi(\overline v) \leftrightarrow \psi (\overline v))$
we have exactly your interpretation.
We can compare it with Dirk van Dalen, Logic and Structure (5th ed - 2013), where there is an explicit "convention" in Definition 3.4.4 [page 67] :
(i) $\mathfrak A \vDash \varphi$ iff $\mathfrak A \vDash Cl(\varphi)$, [where $Cl(\varphi)$ is the universal closure of $\varphi$]
[...]
(iv) $\Gamma \vDash \varphi$ iff $(\mathfrak A \vDash \Gamma ⇒ \mathfrak A \vDash \varphi)$, where $\Gamma \cup \{ \varphi \}$ consists of sentences.
See page 98 :
Definition 4.1.4 : A theory $T$ is a collection of sentences with the property $T \vdash \varphi ⇒ \varphi \in T$.
Finally, see the brief treatment of quantifier elimination [page 121-on] :
Since $T$ admits quantifier elimination, there is a quantifier-free $\psi(x_1,\ldots, x_n)$ such that $T \vdash \varphi \leftrightarrow \psi$.
Addendum
Note that in Marker's book intro [page 4] there is a reference to Joseph Shoenfield, Mathematical Logic (1968) as a "background in mathematical logic".
In Shoenfield's book [page 83] we have :
We say that [a formula] $A$ is equivalent to $B$ in [a theory] $T$ if $\vdash_T A \leftrightarrow B$. We say that $T$ admits elimination of quantifiers if every formula in $T$ is equivalent in $T$ to an open formula.
2nd Addendum
Regarding C.C.Chang & H.Jerome Keisler, Model Theory (3rd ed, 1990), they are in the "mainstream" tradition of model theory, "restricting" the basic semantics definition of theory and logical consequence to sentences.
But [see page 50] they introduce the topic of quantifier elimination in this way :
we introduced the notion of a sentence $\varphi$ being a consequence of a set
$\Sigma$ of sentences, in symbols $\Sigma \vDash \varphi$. What meaning shall we give to $\Sigma \vDash \varphi$ if $\varphi$ is a formula? We shall say that a formula $\varphi (v_0 \ldots v_n)$ is a consequence of $\Sigma$, symbolically $\Sigma \vDash \varphi$, iff for every model $\mathfrak A$ of $\Sigma$ and every sequence $a_0,\ldots, a_n \in A$, $a_0,\ldots, a_n$ satisfies $\varphi$. It follows that the formula $\varphi(v_0 \ldots v_n)$ is a consequence of $\Sigma$ if and only if the sentence $(\forall v_0 \ldots v_n) \varphi(v_0 \ldots v_n)$ is a consequence of $\Sigma$.
We say that two formulas $\varphi, \psi$ are $\Sigma$-equivalent iff $\Sigma \vDash \varphi \leftrightarrow \psi$.
Thus, they prove [page 52] :
THEOREM 1.5.3. Every formula $\varphi$ is $\Delta$-equivalent to an open formula $\psi$ [where $\Delta$ is the theory of dense simple order without endpoints].