We investigate how various treatments of exact exchange affect defect charge transition levels and band edges in hybrid functional schemes for a variety of systems. We distinguish the effects of long-range vs short-range exchange and of local vs nonlocal exchange. This is achieved by the consideration of a set of four functionals, which comprise the semilocal Perdew-Burke-Ernzerhof (PBE) functional, the PBE hybrid (PBE0), the Heyd-Scuseria-Ernzerhof (HSE) functional, and a hybrid derived from PBE0 in which the Coulomb kernel in the exact exchange term is screened as in the HSE functional but which, unlike HSE, does not include a local expression compensating for the loss of the long-range exchange. We find that defect levels in PBE0 and in HSE almost coincide when aligned with respect to a common reference potential, due to the close total-energy differences in the two schemes. At variance, the HSE band edges determined within the same alignment scheme are found to shift significantly with respect to the PBE0 ones: the occupied and the unoccupied states undergo shifts of about +0.4 eV and -0.4 eV, respectively. These shifts are found to vary little among the materials considered. Through a rationale based on the behavior of local and nonlocal long-range exchange, this conclusion is generalized beyond the class of materials used in this study. Finally, we explicitly address the practice of tuning the band gap by adapting the fraction of exact exchange incorporated in the functional. When PBE0-like and HSE-like functionals are tuned to yield identical band gaps, their respective results for the positions of defect levels within the band gap and for the band alignments at interfaces are found to be very close.