I think you're misinterpreting what that potential is that you showed. It's not a real potential. It comes from transforming the problem into a rotating coordinate system and it is velocity dependent. The spacecraft is not in danger of falling down the hill to infinity if it goes outside the L2 point.
I know that, I was mostly trying to express the directionality. In my head the frame of reference is the Earth-Sun axis. Perhaps I should refer to the directions as nadir and zenith. Haven't looked into it but my inclination is to think a fall to zenith from L2 would end up in a heliocentric orbit with approximately 1 year period, and a fall to nadir would do the same, subject to possible encounters with Earth or the Moon. I am guessing the fall to nadir would be quite sensitive to the precise starting conditions and could hit Earth, the Moon, or be ejected from the Earth-Moon system into a heliocentric orbit. Any of the above heliocentric orbits would likely be pretty close to Earth's orbit around the sun, hence have the possibility of future encounters.
JWST has a stationkeeping problem with several constraints:
-It, apparently, cannot thrust or cannot thrust very much in the nadir direction.
-While not tending to fall to infinity, an object at L2 is, as far as I can tell, stable in the directions normal to the Earth-Sun axis, and unstable along the Earth-Sun axis.
-Stationkeeping propellant is limited so the most efficient stationkeeping position compatible with other constraints must be used.
This to me suggests JWST must remain slightly to nadir of L2, even though this is a less efficient use of stationkeeping propellant. It would then maintain its position with delta-v to zenith, which it can do, rather than delta-v to nadir, which it cannot do (since nadir is always aligned to the Earth-Sun axis and JWST cannot expose its instruments to direct sunlight.
The tradeoff I describe is that the closer they get to L2 the more efficient stationkeeping will be, so there are competing requirements: efficient use of propellant, and not overshooting L2 to zenith.
I believe there's also an elliptical orbit about L2 on a plane normal to the Earth-Sun axis, though there's no problem using thrusters in that direction so that's less of an issue. And, with my understanding of the gravitational potentials at work, the spacecraft would tend to move back towards L2 rather than away in that plane.
