Your code is doing exactly what it is supposed to be doing....run when the switch is not set (i.e. pressure is below the threshold). It just so happens that the pressure switch has a cutoff of ~115 PSI. There's a discussion of this
here.
If you really want to get your system pressure to 120 PSI, you can run the pump for a fixed number of cycles after the switch was set (i.e. pressure is above threshold) before giving it the command to shut off. We've used this method for the past few years, and it works just fine.