We study the question of universality in the two-dimensional spin-1 Baxter-Wu model in the presence of a crystal field ∆. We employ extensive numerical simulations of two types, providing us with complementary results: Wang-Landau sampling at fixed values of ∆ and a parallelized variant of the multicanonical approach performed at constant temperature T . A detailed finite-size scaling analysis in the regime of second-order phase transitions in the (∆, T ) phase diagram indicates that the transition belongs to the universality class of the 4-state Potts model. Previous controversies with respect to the nature of the transition are discussed and possibly attributed to the presence of strong finite-size effects, especially as one approaches the pentacritical point of the model.