||(some explanations related to the first post are in edit)
Did you even read my post ? :)
It doesn't work if we want the minimum set of edges with smallest index.
Well, do we need it? :) I mean, is there any more or less practical problem that asks it?
Here is a tough case: Source and Sink are both connected to a complete graph K_n, all egdes have the same capacity.
Consider any decomposition of the vertices of K_n into sets of != 2 vertices. Take any set S with |S| >= 3 and form a cycle with its vertices. In the 'obvious' minimum cut there are only edges (Source, v) and (v, Sink) but it is clear that with such cycle formed you can take any(but only one at a time) vertice v from it and change the edge (Source, v) to (v, next[v]). This minimum cut will contain the same number of edges and you can play this trick with any vertice in any set of any decomposition. The number of such 'min cuts' can be calculated, but even now we can say it's large enough.