There are a lot of subtleties here. For technical reasons, I'll use $\mathsf{WKL}_0$ instead of $\mathsf{RCA}_0$ to explain them.
The following two theorems are found in Simpson's book:
Theorem VIII.2.6. The following is provable in $\mathsf{WKL}_0$. For all $X \subseteq \mathbb{N}$, there exists a countable coded strict $\beta$-model $M$ such that $X \in M$.
Theorem VIII.2.11. The following is provable in $\mathsf{ACA}_0$. For all $X \subseteq \mathbb{N}$, there exists a countable $\omega$-model $M$ of $\mathsf{WKL}_0$ such that $X \in M$.
By Theorem VIII.2.2 countable coded strict $\beta$-model is a countable coded $\omega$-model which satisfies $\mathsf{WKL}_0$. In Theorem VIII.2.6, it is true that the $\omega$-model satisfies $\mathsf{WKL}_0$ but this is not provable in $\mathsf{WKL}_0$. Indeed, $\mathsf{WKL}_0$ is too weak to properly make sense of "satisfies" in this context.
Because of the subtleties I outlined in my blog post (see also this MO answer by Carl Mummert), the statement "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}_0$" has two possible meanings. This leads to a conundrum:
If you interpret "satisfies" using the sort of translation as I outline at the end of my blog post, "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}_0$" is actually equivalent to $\mathsf{WKL}_0$.
If you interpret "satisfies" using valuations as in Simpson's book, then the statement "every set is contained in a countable coded $\omega$-model of $\mathsf{WKL}_0$" is equivalent to $\mathsf{ACA}_0$.
The reason for this solely depends on the meaning of "satisfies" and not on the existence of the models in question. So please be careful when saying that $\mathsf{ACA}_0$ is equivalent to "every set is contained in an $\omega$-model of $\mathsf{RCA}_0$" (and variants).
That said, to address your reference request, Simpson writes in the notes to VIII.2:
Theorem VIII.2.11 and lemma VIII.2.15 are well known, but their origins seem difficult to trace. See the references in Shoenfield [220], e.g., Kleene [142, §72].
These references are:
- J. R. Shoenfield, Degrees of models, J. Symbolic Logic 25 (1960), 233–237.
- S. C. Kleene, Introduction to Metamathematics, Van Nostrand, 1952.
It is also unclear to me how to correctly attribute the result you have in mind.
