Two embedded smooth surfaces in a 4-manifold are an exotic pair if they are topologically, but not smoothly, isotopic A subtle point is that such surfaces might be still equivalent, i.e., related by a diffeomorphism. The first examples of this phenomenon are due to Baraglia (2024), using fairly large 4-manifolds. We present two sorts of examples, in much smaller manifolds. We give examples of equivalent exotic closed surfaces in 2(S^2 x S^2), that carry a non-trivial homology class. We also give examples of equivalent exotic disks in a punctured S^2 x S^2 in the trivial (relative) homology class. In both settings the surfaces are not related by a diffeomorphism that acts by the identity on homology, the first such examples. This is joint work with Dave Auckly, Hokuto Konno, Anubhav Mukherjee, and Masaki Taniguchi.