It’s been a long time since I got my astronomy degree, but your version is what I recall also. Whatever small rotational perturbation in the initial gas becomes more pronounced as it coalesces in on itself and defines the plane of the star system. Planets form within this plane after it is defined, and they all travel in the same direction around the star.

Regarding galaxies, the most common spiral ones like our own Milky Way follow the same principle at a larger scale. But there are also elliptical galaxies, not to mention irregular ones. In an elliptical galaxy, there is a more random movement of stars in a cloud around its core. So they look more 3D I guess, to go back to what the OP was asking about. I seem to recall the most accepted explanation for how these form is from the aftermath of a collision between 2 spirals? So presumably, when our galaxy collides with Andromeda in several billion years time, the resulting combined galaxy may emerge as an elliptical?