The model-theoretic definition of logic quoted in the question is given in that book to be used in Lindström's theorem. Nevertheless, equational logic is a model-theoretic logic and the induced consequence relation is equivalent to the one defined via proof theory (for this logic is complete).

Now, other questions about other general definition of logic were raised in the comments. In order to address those questions we need some preliminary remarks and notation.

Let $X$ be a set (think of $X$ as a set of formulas). A logic on $X$ is a subset of $\wp(X)\times X$ (think of such a subset as a consequence relation). We use $\Gamma$ and $\Delta$ to denote general subsets of $X$ and $\phi$ and $\psi$ to denote general elements of $X$.

A valuation on $X$ is a subset of $X$ (think of such a subset as the formulas which are true according to the valuation).

We say that a valuation $w$ is compatible with a logic $l$ iff for every $(\Gamma,\phi)\in l$, if $\Gamma\subseteq w$, then $\phi\in w$.
If $l$ is a logic on $X$, then $l$ defines a set of valuations $G(l)$, those compatible with $l$.

Conversely, if $m$ is a set of valuations, then $m$ defines a logic $L(m)$: the set of the pairs $(\Gamma,\phi)$ such that for all $w\in m$, if $\Gamma\subseteq w$, then $\phi\in w$.
$G$ and $L$ constitute a Galois connection.

For a given logic $l$, the following are equivalent:

- There is a set of valuations $m$ such that $l=L(m)$
- $l=L(G(l))$
- $l$ is reflexive, idempotent and monotonic.

A logic $l$ is reflexive if for every $\phi\in\Gamma$, $(\Gamma,\phi)\in l$.
A logic $l$ is idempotent if whenever $\Delta, \Gamma\subseteq X$, $\phi\in X$, $(\Gamma, \phi)\in l$ and for every $\psi\in\Gamma$ , $(\Delta,\psi)\in l$, we have $(\Delta,\phi)\in l$. A logic $l$ is monotonic if whenever $\Gamma\subseteq\Delta$, if $(\Gamma,\phi)\in l$, then $(\Delta,\phi)\in l$.

Now, we can consider the question:

Is the reverse true: does $\models$ between sets of formulas and formulas induce $\models$ between structures and formulas?

Yes, in some trivial sense: We can look at the structures inducing valuations compatible with $l$ (assuming that structures induce valuations in that case). But the difficulty is to recover your first consequence relation from the induced relation. This is not always possible:

We can look at the structures inducing valuations compatible with $l$, then we can look at all the consequence relations preserved by those structures. In general, we will not recover $l$ this way, even if $l$ is a finitary, reflexive, idempotent and monotonic relation. We can start with a set $m$ of valuations which are not induced by structures. We can take $l=L(m)$, and this will be finitary if $X$ is finite, for example.

But if we allow those general valuations, not only those induced by structures, then yes, from a tarskian logic $l$ we can define a canonical set of valuations $G(l)$ such that $l$ can be recovered from that set.