We investigate the correlation between Disturbance Storm Time (Dst) characteristics and solar wind conditions for the main phase of geomagnetic storms, seeking possible factors that distinguish extreme storms (minimum Dst <−250 nT) and major storms (minimum Dst <−100 nT). In our analysis of 170 storms, there is a marked correlation between the average rate of change of Dst during a storm’s main phase (ΔDst/Δt) and the storm’s minimum Dst, indicating a faster ΔDst/Δt as storm intensity increases. Extreme events add a new regime to ΔDst/Δt, the hourly time derivative of Dst (dDst/dt), and sustained periods of large amplitudes for southward interplanetary magnetic field Bz and solar wind convection electric field Ey. We find that Ey is a less efficient driver of dDst/dt for extreme storms compared to major storms, even after incorporating the effects of solar wind pressure and ring current decay. When minimum Dst is correlated with minimum Bz, we observe a similar divergence, with extreme storms tending to have more negative Dst than the trend predicted on the basis of major storms. Our results enable further improvements in existing models for storm predictions, including extreme events, based on interplanetary measurements.