This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).

Your $S(N)$ is naturally a scheme over $\mathbb{Z}[1/N, \zeta_N]$. Your $X_{\widetilde{\Gamma}(N)}$ is, as you have defined it, a scheme over $\mathbb{Z}[1/N]$. But there is actually a natural map $\mathbb{Z}[1/N, \zeta_N] \hookrightarrow \mathcal{O}(X_{\widetilde{\Gamma}(N)})$ which sends $\zeta_N$ to $e_N(P, Q)$; and if you regard $X_{\widetilde{\Gamma}(N)}$ as a $\mathbb{Z}[1/N, \zeta_N]$ via this morphism, it coincides with $S(N)$.

This is a subtle issue (which has come up before on this site several times, see e.g. is the modular curve X(N) defined over Q? for a related question).

Your $S(N)$ is naturally a scheme over $\mathbb{Z}[1/N, \zeta_N]$. Your $X_{\widetilde{\Gamma}(N)}$ is, as you have defined it, a scheme over $\mathbb{Z}[1/N]$. But there is actually a natural map $\mathbb{Z}[1/N, \zeta_N] \hookrightarrow \mathcal{O}(X_{\widetilde{\Gamma}(N)})$ which sends $\zeta_N$ to $e_N(P, Q)$; and if you regard $X_{\widetilde{\Gamma}(N)}$ as a $\mathbb{Z}[1/N, \zeta_N]$-scheme via this morphism, it coincides with $S(N)$.