A matching cut is a matching that is also an edge cut. In the problem Minimum Matching Cut, we ask for a matching cut with the minimum number of edges in the matching. We investigate the differences in complexity between Minimum Matching Cut, its counterpart Maximum Matching Cut, and the decision problem Matching Cut. Our polynomial-time algorithms for $P_8$-free, $S_{1,1,3}$-free and $(P_6 + P_4)$-free graphs extend the cases where Minimum Matching Cut and Maximum Matching Cut are known to differ in complexity. In addition, they solve open cases for the well-studied problem Matching Cut. The NP-hardness proof for $3P_3$-free graphs implies that Minimum Matching Cut and Matching Cut, which is polynomial-time solvable even for $sP_3$-free graphs, for any $s \geq 1$, differ in complexity on certain graph classes. Further, we give complexity dichotomies for both general and bipartite graphs of bounded radius and diameter.