A reformulated version of Grimme's most recent DFT dispersion correction with Becke-Johnson damping (DFT-D3(BJ)) is presented, which only depends on C6 dispersion coefficients. The role of the higher order correction terms in the DFT-D3(BJ) model is critically investigated, and a sigmoidal interpolation function for adjusting to different density functional approximations (DFA) is employed alternatively, while keeping finite damping of Becke and Johnson. For the proposed C6-only dispersion correction scheme, only one parameter needs to be fitted per DFA (instead of three for DFT-D3(BJ)). Eight standard DFAs from different classes are parametrized and evaluated. In comparison to DFT-D3(BJ), one of the most accurate corrections up to date, the new correction shows only negligible deviations in accuracy for the huge GMTKN30 benchmark set.