Boiled down, you done just right I’d say.
The correct response to an alarm indicating an intruder is high alert, condition one.
If my alarm goes off for a fire a lady’s voice starts saying, “Fire, Fire, Fire”. She never has said, “Intruder, Intruder, Intruder” yet.
Just like “My Cousin Vinny” when Pesci and lovely Marisa are staying in the log cabin and the owl shrieks! He comes out ablazin!! LOL!