In my discrete math class, our professor is insistent on using a labeling algorithm to find the max flow of a graph. This is fine, although the Ford-Fulkerson algorithm makes more sense to me. When the labeling algorithm terminates, it has the side-effect of giving us sets $P$ and $\bar{P}$ separated by a minimum cut based on which vertices have been labelled when the algorithm terminates. My understanding is that the Ford-Fulkerson algorithm won't provide a specific minimum cut, but will give you the min-cut value.
Are all minimum cuts equivalent, at least in these small introductory problems?
Extra question: I can't even begin to think of how I'd make this rigorous, but would it be reasonable to say that the final residual graph given by the Ford-Fulkerson algorithm could give us minimum cut sets $P$ and $\bar{P}$ based on where the path from $s$ to $t$ breaks down on the saturated edges?
I'm not sure what you mean by "equivalent", but there may be many different minimum cuts. As an extreme example, if your graph consists of a source $s$, sink $t$, and $n$ other vertices each having an edge from $s$ of weight $1$ and an edge to $t$ of weight $1$, then all of the $2^n$ possible cuts are minimum cuts of value $n$. But all minimum cuts have the same value, by definition.