In Kaye's "Models of Peano Arithmetic" in chapter 9 on satisfaction a lot of effort is expanded to prove that $\text{Sat}_{\Delta_0}(x, y)$, representing truth of $\Delta_0$-sentences, is a $\Delta_1$-sentence under $\textbf{PA}$ (Theorem 9.13 in Kaye, p. 125). However, I'm unsure how this is used. Here is the theorem:
Kaye then concludes:
I guess the latter is a result of the properties listed in Theorem 9.13 plus induction in the metatheory on the complexity of formulas. I presume it is here that $\text{Sat}_{\Delta_0}(x, y)$ is $\Delta_1(\textbf{PA})$ is used, because somewhat earlier in the chapter Kaye has a similar strategy. He proves:
And then mentions a similar result where instead the $\Delta_1$-ness of the formula $\text{term}(x)$ is used.
However, I also don't see how $\mathbb{N} \models \text{term}(m) \Leftrightarrow \textbf{PA} \models \text{term}(m)$ is used here. Furthermore, isn't it enough that $\text{term}(x)$ is $\Sigma_1$ to show this?
FYI there seems to be a typo in Kaye: in the claimed ${\bf PA}$-theorem scheme of the "in particular" part, the first block of $v$s should be a block of $x$s.
I don't think the $\Delta_1$-ness of $\mathsf{Sat}_{\Delta_0}$ is in fact used in the "in particular" passage. Rather, Theorem $9.13$ proves two things:
a complexity bound on $\mathsf{Sat}_{\Delta_0}$, modulo ${\bf PA}$; and
some nice ${\bf PA}$-provable properties of $\mathsf{Sat}_{\Delta_0}$.
The "in particular" refers only to this second point; at a glance, the metatheoretic induction taking place (your guess is correct re: this) only uses the fact that ${\bf PA}$ proves enough to "break down" instances of $\mathsf{Sat}_{\Delta_0}$. Now granted, this second bulletpoint is vastly simpler than the first and, as its proof indicates, it (and consequently the "in particular" bit) could have been stated right after the Explanation of the Definition of $\mathsf{Sat_{\Delta_0}}$ in the first place.
But nonetheless the $\Delta_1$-ness of the formula $\mathsf{Sat}_{\Delta_0}$ (as always, modulo ${\bf PA}$) is interesting in its own right. In general, keeping track of the "provable complexity" of a formula is a useful thing to do. Note that this type of analysis is not subsumed by the computability-theoretic interpretation of the arithmetic hierarchy: while knowing that $\mathsf{Sat}_{\Delta_0}$ is computable is enough to conclude that it is $\Delta_1$ in the structure $\mathbb{N}$, that is not enough to conclude that it is $\Delta_1$ modulo ${\bf PA}$. So this sort of analysis is nontrivial.
Meanwhile, as to your last question, note that the $\Sigma_1$-ness of $\mathsf{term}(x)$ gives us $$\mathbb{N}\models\mathsf{term}(m)\implies{\bf PA}\models\mathsf{term}(m)$$ (conflating a number with its numeral, as usual). However, for the converse implication we need that $\mathsf{term}(x)$ is $\Pi_1$. For example, take $\mathsf{term}(x)$ to be "${\bf PA}$ is inconsistent" (so it ignores $x$ completely). Then - unless we grant more than mere consistency of ${\bf PA}$ - it could be the case that ${\bf PA}\models \mathsf{term}(0)$ (say) while $\mathbb{N}\not\models\mathsf{term}(0)$.