I'm not sure how much this will add, but the points are exactly that - just points. Gravitational bodies are often reduced to points for the purpose of calculations, such as, centers of mass and barycenters are also points. The Lagrange points are the same - mathematically calculated points based on the two objects in orbit around their mutual barycenter....
From this source I get:
The size of these islands varies. Each planet in the solar system has
its own Lagrangian points. The islands of stability get bigger farther
from the Sun and also for more massive planets. The ones associated
with Earth are roughly 500,000 miles (800,000 kilometers) wide. The
biggest zones (at least in the solar system) ...