The stop-signal task (SST) is widely used for studying the speed of the latent process of response inhibition. The SST patterns are typically explained by a horse-race model (HRM) with supposed Go and Stop processes. However, HRM does not agree with the sequential-stage model of response control. As a result, the exact relationship between the response selection, the response execution stages, and the Stop process remains unclear. We propose that response selection occurs within the stop-signal delay (SSD) period, and that the competition between the Go and Stop processes occurs within the response execution period. To confirm this, we conducted two experiments. In Experiment 1, participants carried out a modified SST task with an additional stimulus category - Cued-Go. In the Cued-Go trials, cues were followed by imperative Go signals. The Cue-Go period duration was dynamically adjusted by an adaptive algorithm based on the response times reflecting the individual response selection duration. In Experiment 2, Cued-Go stimuli were followed by Stop Signals in half of the trials and response inhibition efficiency was calculated. The results of Experiment 1 indicate that SSD reflects the duration of the response selection process. The results of Experiment 2 show that this process has an independent and small effect on the effectiveness of controlled inhibition of the target response. Based on our findings, we propose a two-stage model of response inhibition in SST, with the first stage including response selection process and the second stage response inhibition following the SS presentation.