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?
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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.