An alarm "goes off" when it rings or sounds an alarm. "To go off" can also mean to fire, or to explode, or to detonate. I don't like the example your dictionary gives for meaning #1. But here are some examples of usage of "to go off" with this meaning. 1) He tried to smoke a cigarette in the men's room, but as soon as he lit it, the fire alarm went off. 2) I have set my alarm clock to go off at 7 a.m. and wake me up in time to get to work. 3) He accidentally dropped his gun and it went off. A gun isn't supposed to go off unless you pull the trigger. 4) Nitroglycerin is dangerous because it can go off easily, from the slightest shock. Unfortunately, that is only one group of meanings. The phrasal verb "to go off" has an amazing number of meanings that can be quite different. For example, it can refer to food spoiling. "Phew. This milk smell bad. It's gone off."

According to the Cambridge dictionary If a warning device goes off, it starts to ring loudly or make a loud noise: "The alarm should go off automatically as soon as smoke is detected." "Didn't you hear your alarm clock going off this morning?"