This article is the third one in a series of papers by the authors on vanishing-viscosity solutions to rate-independent damage systems. While in the first two papers (Knees, D. et al. 2013 Math. Models Methods Appl. Sci.23(4), 565–616; Knees, D. et al. 2015 Nonlinear Anal. Real World Appl.24, 126–162) the assumptions on the spatial domain Ω were kept as general as possible (i.e., non-smooth domain with mixed boundary conditions), we assume here that ∂Ω is smooth and that the type of boundary conditions does not change. This smoother setting allows us to derive enhanced regularity spatial properties both for the displacement and damage fields. Thus, we are in a position to work with a stronger solution notion at the level of the viscous approximating system. The vanishing-viscosity analysis then leads us to obtain the existence of a stronger solution concept for the rate-independent limit system. Furthermore, in comparison to [18, 19], in our vanishing-viscosity analysis we do not switch to an artificial arc-length parameterization of the trajectories but we stay with the true physical time. The resulting concept of Balanced Viscosity solution to the rate-independent damage system thus encodes a more explicit characterization of the system behaviour at time discontinuities of the solution.