SLC Tower Controller reported an air carrier which had been instructed to taxi into position and hold began their takeoff roll with another air carrier taxiing across the runway.

Narrative

I was working Local Control Center. Semi busy with the morning bank and deicing. I just got on position. After Aircraft X landed Runway 34R; I put Aircraft Y in position and told him that 'traffic will cross down field.' I used my own technique of strip management indicating that I put an aircraft in position; and I used the yellow Runway Crossing memory aid over Aircraft Y's strip on my board. I gave exit instructions to Aircraft X and gave further instructions. Aircraft X was a FS; Taxi back and needed to pick up his IFR clearance. Aircraft X is not involved in this situation. I then told Aircraft Z 'traffic holding in position; Cross Runway 34R at S'; join hotel then contact ground. When Aircraft Z was about half way done crossing the runway; I observed Aircraft Y departing and the green lights on the runway lit up; but no voice. I told Aircraft Y to 'Cancel Departure Clearance. STOP.' Aircraft Y successfully responded to my cancel instructions; and said that he 'saw the traffic' downfield crossing. I exited Aircraft Y off the runway and switched him to GC.