The passage at Earth of the October 1995 magnetic cloud and the high‐speed corotating stream overtaking it, monitored by the Global Geospace Science (GGS) spacecraft Wind, caused two consecutive geomagnetic storms: a major one during the strong Bz < 0 nT phase of cloud passage and a moderate one during the intermittent Bz < 0 activity in the fast corotating stream. Large dynamic pressure changes were observed in the sheath region ahead of the cloud and in the cloud‐stream interface region at its rear, resulting in substantial corrections to the measured Dst index. A burst of superdense plasma sheet extending over ∼2 hours in local time was observed at geostationary orbit during the second storm. We simulate the ring current development during this storm period using our kinetic model and calculate the magnetic field perturbation caused by the ring current. The plasma inflow on the nightside is modeled throughout the investigated period using data measured at geosynchronous orbit. The modeled Dst index is compared with the observed Dst values corrected for magnetopause and telluric currents. The temporal evolution of the ring current H+ and O+ distribution functions is computed, considering losses due to charge exchange, Coulomb collisions, and ion precipitation. We find that (1) the storm time enhancement of the plasma sheet ion population contributed significantly to the ring current buildup; (2) an additional ∼12 nT decrease in Dst is achieved when the symmetry line of the plasma convection paths is rotated eastward from the dawn‐dusk direction with 3 hours during the first storm; (3) the major loss process is charge exchange, followed by Coulomb collisions and ion precipitation; (4) however, the energy losses due to ion precipitation increase monotonically during the more active periods, reaching the level of Coulomb losses at peak storm intensity. We argue that the losses due to ion precipitation considered in this study are closely related to the enhanced convection electric field, which in our model is parameterized with the planetary Kp index. Correspondingly, we find that (5) there is a very good correlation between the variations in time of this index and the magnitude of the ion precipitation losses.