I have seen this post, where they identify symplectic structures on a compact orientable surface with $\mathbb R\smallsetminus0$, which I understand except for one issue regarding the orientation on $\Sigma$.
Let $\Sigma$ be a closed orientable surface surface, and let $\omega$ and $\omega'$ be symplectic forms on $\Sigma$. Then each $\omega$ and $\omega'$ determines an orientation on $\Sigma$, and so any diffeomorphism such that $F^*\omega'=\omega$ preserves orientation. However, suppose that $\omega$ and $\omega'$ induce different orientations, then $F:\Sigma\rightarrow \Sigma$ is still an orientation preserving map, but I feel like the claim that $[\omega]=[\omega']$ no longer holds. In particular if we denote by $\Sigma$ the surface with the orientation induced by $\omega$, and $\Sigma'$ the surface with the orientation induced by $\omega'$, then it is true that: $$\int_\Sigma \omega=\int_{\Sigma'}\omega'$$ however, if the orientations disagree then: $$\int_\Sigma\omega'=-\int_\Sigma \omega$$ so $[\omega]=-[\omega']$ in $H^2(\mathbb R)$. In particular, if $\Sigma=\mathbb S^2$, there is an antipodal map $t$, so there exists an example of this form with $(\mathbb S^2,\omega)$ and $(\mathbb S^2,-\omega)$.
Moreover, having read Moser's original paper, he proves his general statement (Moser's theorem) on a closed manifold $M$ for orientation preserving automorphisms, which I assume means a fixed orientation on $M$, as if he doesn't Moser's trick completely falls apart. So I am unsure of how to reconcile this difference in the other direction...