[ARCHIVE ANSWER]
Police cannot check for warrants or run a criminal history based on a license plate check. A license plate check contains the name of the registered owner, their address, and a description of the car.
To check for warrants or criminal history, an officer must have a name, race, gender, and date of birth. Dates of birth are not in license plate records, so the cop must stop you to check your criminal history and to check for warrants.
If you were ticketed for the pot instead of placed in jail, the violation will not appear on your criminal history.