I think you are confused about stability. It does not matter about the nature of the gravitational field, what matters is whether the resulting orbits are relatively stable or relatively unstable. That is, how likely they are to remain close to those points.
It's a bit like being in a hill or a valley. In a valley you're more likely to remain in the valley - it's harder to get out and easy to stay in. On the top of a hill it's easy to roll off and hard to stay up. One is stable place to be, the other is not a stable place to be.
The fact that spacetime distorted by mass and energy is how gravity arises does not change this. Spacetime is not the thing that is stable or unstable, it is the nature of the orbit the objects are in that either makes it more likely or less likely to stay near a particular Lagrange point.