I'm not sure if this formulation with $\omega$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).

If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:

Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $\phi(\bar{v},w)$, if $M,N\models T$, $A$ is a common substructure of $M$ and $N$, $\bar{a}\in A$, and there is $b\in M$ such that $M\models\phi(\bar{a},b)$, then there is $c\in N$ such that $N\models \phi(\bar{a},c)$. Then $T$ has quantifier elimination.

It is not too difficult to deduce the theorem you quoted from this.

I'm not sure if this formulation with $\omega_1$-saturated models is in any of the "main standard" texts. But there is a complete treatment in these course notes by Pillay (see Proposition 2.29).

If you're looking for something in a standard published text, Corollary 3.1.6 of Model Theory: An Introduction (Marker) is along the same lines. It says:

Let $T$ be an $L$-theory. Suppose that for all quantifier-free formulas $\phi(\bar{v},w)$, if $M,N\models T$, $A$ is a common substructure of $M$ and $N$, $\bar{a}\in A$, and there is $b\in M$ such that $M\models\phi(\bar{a},b)$, then there is $c\in N$ such that $N\models \phi(\bar{a},c)$. Then $T$ has quantifier elimination.

It is not too difficult to deduce the theorem you quoted from this.