To find the time that the sample spends in stationary phase (adjusted retention time), we just need to subtract the time, in which it is in the column but not in stationary phase.
This is a misconception (a subtle point here). The adjusted retention time is the time an analyte spends in the column not the stationary phase. Remember that a chromatography column is made up of a stationary phase plus the mobile phase. The sample continuously partitions between the mobile phase and the stationary phase.
Dead time simply refers to the time it takes for a sample to travel from the injector to the detector without having any interaction with the stationary phase. Imagine a garden hose, when you open the tap, it take a while for the water to reach the other end. Dead time is exactly like that.
Calculating the fraction of the time spent by an analyte in the stationary phase is slightly more involved. Check this link on "Migration rate of solutes in a chromatographic separation"