The idea of plainly stable and unstable points only happen if you have rotational symmetry, and doesn't really apply to most of the points. Leading the discussion on those terms isn't productive.
It orbites in a plane where it's always subject to attraction, but that doesn't mean the point always attracts the satellite.
https://en.wikipedia.org/wiki/File:Lagrange_points2.svg
The idea of plainly stable and unstable points only happen if you have rotational symmetry, and doesn't really apply to most of the points. Leading the discussion on those terms isn't productive.
It orbites in a plane where it's always subject to attraction, but that doesn't mean the point always attracts the satellite.