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 $$ \iota_{X_H} \omega = - dH. $$

We want to choose the Liouville form $\lambda$ on the Weinstein domain $W$, so that $\lambda |_{\d W} = \alpha$. Here $W = \{ |z| \leq 1\}$.

So, we can use $\lambda = -y dx + x dy$ (which restricts to $d\theta$).

Given the above sign convention, we want $$ \lambda = -y dx, \quad \omega = dx \wedge dy $$ and Reeb flow is the negative geodesic flow on the boundary.