For each algebra of global dimension 2 arising from the quiver with potential associated to a triangulation of an unpunctured surface, Amiot-Grimeland have defined an integer-valued function on the first singular homology group of the surface, and have proved that two such algebras of global dimension 2 are derived equivalent precisely when there exists an automorphism of the surface that makes their associated functions coincide. In the present paper we generalize the constructions and results of Amiot-Grimeland to the setting of arbitrarily punctured surfaces. As an application, we show that there always is a derived equivalence between any two algebras of global dimension 2 arising from the quivers with potential of (valency at least 2) triangulations of arbitrarily punctured polygons. While in the unpunctured case the quiver with potential of any triangulation admits cuts yielding algebras of global dimension at most 2, in the case of punctured surfaces the QPs of some triangulations do not admit cuts, and even when they do, the global dimension of the corresponding degree-0 algebra may exceed 2. In this paper we give a combinatorial characterization of each of these two situations.