The dynamics of elastic interfaces is a general field of interest in statistical physics, where magnetic domain wall has served as a prototypical example. Domain wall 'creep' under the action of sub-threshold driving forces with thermal activation is known to be described by a scaling law with a certain universality class, which represents the mechanism of the interaction of domain walls with the applied forces over the disorder of the system. Here we show different universality classes depending on the driving forces, magnetic field or spin-polarized current, in a metallic system, which have hitherto been seen only in a magnetic semiconductor. We reveal that an adiabatic spin-transfer torque plays a major role in determining the universality class of current-induced creep, which does not depend on the intricacies of material disorder. Our results shed light on the physics of the creep motion of domain walls and other elastic systems.