Geomagnetic storms are primarily driven by stream interaction regions (SIRs) and coronal mass ejections (CMEs). Since SIR and CME storms have different solar wind and magnetic field characteristics, the magnetospheric response may vary accordingly. Using FAST/TEAMS data, we investigate the variation of ionospheric O+ and H+ outflow as a function of geomagnetic storm phase during SIR and CME magnetic storms. The effects of storm size and solar EUV flux, including solar cycle and seasonal effects, on storm time ionospheric outflow, are also investigated. The results show that for both CME and SIR storms, the O+ and H+ fluence peaks during the main phase, and then declines in the recovery phase. However, for CME storms, there is also significant increase during the initial phase. Because the outflow starts during the initial phase in CME storms, there is time for the O+ to reach the plasma sheet before the start of the main phase. Since plasma is convected into the ring current from the plasma sheet during the main phase, this may explain why more O+ is observed in the ring current during CME storms than during SIR storms. We also find that outflow fluence is higher for large storms than moderate storms and is higher during solar maximum than solar minimum.