The Perdew-Zunger (PZ) self-interaction correction (SIC) was designed to correct the one-electron limit of any approximate density functional for the exchange-correlation (xc) energy, while yielding no correction to the exact functional. Unfortunately, it spoils the slowly varying (in space) limits of the uncorrected approximate functionals, where those functionals are right by construction. The right limits can be restored by locally scaling down the energy density of the PZ SIC in many-electron regions, but then a spurious correction to the exact functional would be found unless the self-Hartree and exact self-xc terms of the PZ SIC energy density were expressed in the same gauge. Only the local density approximation satisfies the same-gauge condition for the energy density, which explains why the recent local-scaling SIC is found here to work excellently for atoms and molecules only with this basic approximation and not with the more advanced generalized gradient approximations (GGAs) and meta-GGAs, which lose the Hartree gauge via simplifying integrations by parts. The transformation of energy density that achieves the Hartree gauge for the exact xc functional can also be applied to approximate functionals. Doing so leads to a simple scaled-down self-interaction correction that is typically much more accurate than PZ SIC in tests for many molecular properties (including equilibrium bond lengths). The present work unambiguously shows that the largest errors of PZ SIC applied to standard functionals at three levels of approximation can be removed by restoring their correct slowly varying density limits. It also confirms the relevance of these limits to atoms and molecules.