Our basic symplectic example is $\C$, where we have $$ \omega = dx \wedge dy, \; J(\d_x) = \d_y \; g(v,v) = \omega(Jv,v) \geq 0. $$

The boundary of $\C$ (at infinity) is $S^1$, where we want the Reeb flow to be $\d_\theta$. That requires $\alpha = d\theta$.

If we want to use function $H=r^2/2$ to generate this flow, then $dH = r dr$, and $X_H = \d_\theta$, $\omega =r dr \wedge d\theta$, hence, we better require $$ \iota_{X_H} \omega = - dH. $$