But here is the key: The angular momentum vector does not have to be in the same direction as the angular velocity vector. I know that sounds crazy, but that's what tensors do.

Second, the z-component of angular momentum is negative. This means the angular momentum vector is pointing in a direction that would appear to be into the screen (from your view).

Direct measurement of a 27-dimensional orbital-angular-momentum state vector. Nat. Commun. 5:3115 doi: 10.1038/ncomms4115 (2014). References.