"Accidental" refers to a failure of the weapon mechanically, or something truly accidental, outside the resonable control of the operator. For example, dropping a gun and it goes off. Or, a safety malfunction, or a worn out sear, or a foreign object trips the trigger.
"Negligent" is where the operator squeezes the trigger on a firearm and it discharges when they didn't mean to fire a shot. Excuses are "I thought it was empty" or 'I didn't realize my finger was on the trigger" as so forth.
Truly accidental discharges are very rare. Probably 99% are "negligent". Also, if you are carrying a vintage single action revolver with a round under the chamber (not safe or recommended) and the hammer is caught and released firing the gun by accident...probably still "negligent" for not following safe practices for that firearm. Room for debate in these situations I guess since you didn't press the trigger...but that's probably .001% of cases.
In Iraq, a guy had a discharge in a clearing barrel with an AK. He removed the mag, checked the chamber, but failed to see the round still on the bolt face that didn't eject and was hard to see because of the cover. The chamber was empty. When he closed the bolt and sqeezed the trigger (supposed to at a clearing barrel) it fired. The purpose of a clearing barrel is to safely take a round when something goes wrong.
You could say "negligent" because he didn't look close enough, but he was given the benefit of a doubt (not fired) because the weapon did malfunction in not ejecting and the design lent itself to problems observing the bolt face. Everyone was told of the incident, to check the bolt face and that future incidents would be treated as NDs and you would be fired because everybody was told.