Number of nonequivalent weight functions on a set of $n$ elements

You are asking for the number $T_n$ of threshold functions on the set $\{-1,1\}^n$. According to https://arxiv.org/pdf/1903.06595.pdf (Section 1.2), Zuev showed that $\log_2T_n\sim n^2$.

It turns out my question is answered in a recent paper by Friedrich Eisenbrand, Christoph Hunkenschröder, Kim-Manuel Klein, Martin Koutecký, Asaf Levin, and Shmuel Onn on the complexity of integer programming. Letting $w \in \mathbb{R}^n$ be the vector of weights, the weight of a subset $S' \subseteq S$ can be written by letting $x_{S'}$ be the $0/1$-characteristic vector of $S'$, so that the weight of $S'$ becomes the inner product $w \cdot x_{S'}$.

Theorem 65 in the cited paper shows that for any linear function $f(x) = w \cdot x$ for $x \in \{0,1\}^n$, there is an equivalent weight function $g(x) = w' \cdot x$ such that $w' \in \{- n(12n)^n, \ldots, n(12n)^n\}$, that is, each weight is an integer whose absolute value is $n \cdot n^{O(n)}$. Hence each weight function has a representation with each of the $n$ weights belonging to this range, giving an upper-bound of at most $(1+2n \cdot n^{O(n)})^n = 2^{O(n^2 \log n)}$ nonequivalent weight functions.