I created this edge by linking the two vertices directly (
Make Edge / Face):
Now I would like to split the quad face below along this loose edge. What is the solution for the general case where the edge crosses multiple coplanar faces?
AFAIK, there is no way to use an edge (without even a plane) to divide a face.
What you need to do is delete the edge (x->edge) and then join the two verts, by selecting them and hitting 'j'.
If you've made a bunch of edges not realizing what you were doing in some other model, you can use mesh->cleanup->delete loose to delete all these edges, but I don't believe there's any way to transform them into cuts.