What is Herbrand interpreration and model?
Herbrand models have special domains. The domain of every Herbrand model for $S$ (which is a set of formulas of $L$) is the Herbrand universe $U_L$ for $L$. The set $U_L$ cannot be $\{0, 1\}$, it has a totally different form, see the definition of Herbrand universe in Chapter 1 “Preliminaries”, §3 “Interpretations and models”. The set $U_L$ can contain $p(a)$, $p(b)$, $p(f(g(a)))$ where $a$ and $b$ are constants of $L$, $p$ is a predicate symbol of $L$, $f$ and $g$ are function symbols of $L$, such things. It cannot contain $0$ or $1$.