We have investigated the effect of key solar wind driving parameters on solar wind-magnetosphere coupling efficiency during sheath and magnetic cloud-driven storms. The particular focus of the study was on the coupling efficiency dependence with Alfvén Mach number (MA). The efficiency has been estimated using the dawn-dusk component of the interplanetary electric field (EY), Newell and Borovsky functions as a proxy for the energy inflow and the polar cap potential (PCN), and auroral electrojet (AE) and SYM-H indices as the measure of the energy output. We have also performed a time delay analysis between the input parameters and the geomagnetic indices. The optimal time lag and smoothing window length depend on the coupling function used and on the solar wind driver. For example, turbulent sheaths are more sensitive to the time shift and the averaging interval than smoother magnetic clouds. The results presented in this study show that the solar wind-magnetosphere coupling efficiency depends strongly on the definition used, and it increases with increasing MA. We demonstrate that the PCN index distinctively shows both a Mach number dependent saturation and a Mach number independent saturation, pointing to the existence of at least two underlying physical mechanisms for the saturation of the index. By contrast, we show that the AE index saturates but that the saturation of this index is independent of the solar wind Mach number. Finally, we find that the SYM-H index does not seem to saturate and that the absence of saturation is independent of the Mach number regime. We highlight the difference between the typical MA conditions during sheath regions and magnetic clouds. The lowest MA values are related to the magnetic clouds. As a consequence, sheaths typically have higher solar wind-magnetosphere coupling efficiencies than magnetic clouds.