I'm struggling with some basic intuition regarding the angular velocity $\vec\omega$ and angular momentum $\vec{L}$ vectors, for any arbitrary motion. Specifically, I'm trying to figure out what the idea is behind their directions. (The idea behind the magnitudes is more clear).
First I'll explain what (I think) I know, and please correct any mistakes.
We know that angular momentum is defined as $\vec{L} = m\vec{r}×\vec{v}$
And angular velocity is defined as $\vec\omega = (\vec{r}\times\vec{v})/r^2$
Regarding angular momentum: by the cross product, I understand that the angular momentum is normal to the plane spanned by $\vec{r}$ and $\vec{v}$. So my question is, what would be the intuition about the direction of this vector? Does it matter for good understanding, or is it kind of an arbitrary artifact of the cross-product?
About angular velocity: this one really gets me. So again, it's clear the angular velocity is perpendicular to both the position and the velocity vectors, and thus a normal to the plane spanned by them.
However - again, please explain the intuition for the direction of the angular velocity.
For circular motion on the xy plane, where the origin is the center of the circle, I understand. $\vec\omega$ will be parallel to the axis of rotation, which the particle rotates around.
However, this understanding seems to break down (?) for other cases. For example, in the case of circular motion on the xy plane - but choosing the origin not on the plane. Maybe in the center of the plane, but elevated some distance above the plane. In this case, $\vec\omega$ is no longer parallel to what seems to be the axis of rotation for the particle (which is still the Z axis). In fact, in this case, $\vec\omega$ changes its direction constantly.
So please explain the intuition, or idea, about the direction of $\vec\omega$ in the general case.
(Note - I have asked a similar question in the past few days, but it had to do with fixed-axis rotation only. This is asking about the general case).