I doubt that normal folks in Gotham are aware of his rule as he never made a public announcement. Even if they heard of it from someone, there are so many incidents where Batman killed someone accidentally that they will always be in doubt.
While Batman stopped killing ever since the 1943 due to becoming a kid friendly character, despite the fact that he got his moral code he resumed occasional killing starting with the late 1960's in Brave and The Bold # 83 in which he destroys a German plane using a hand grenade and uses dynamite to blow up a convoy of German soldiers as they are crossing a bridge. As Batman said while breaking his rule in Batman #420 (1988): Sometimes you have to ignore the rules. I'm not in this business to protect the rules, I serve justice. Since there's also too many instances showing Batman occasionally killing since the late 60's, here are just a few more examples:
Detective Comics #29
Detective Comics #33. He throws a vial of sleeping gas at the pilot who crashes into a river
There are a lot more examples like that in comics.
In Nolan's trilogy itself total deaths caused by Batman is approximately 10:
In Batman Begins, the explosive chain reaction that Bruce started caused many deaths (8 onscreen) but they most likely weren't intended. Bruce wanted to have a distraction to escape and weaken the League of Shadow ninjas, but the fire caused more than he counted on.
By the end of the movie Batman lets Ra's Al Ghul die by leaving him in a racing train whose controls has been disabled
In The Dark Knight Batman saves Joker from certain death but soon after kills Two-Face by throwing him and himself off the edge.
This is apparently an unplanned kill as well. Two-Face was standing close to the edge and Batman jumped at him trying to stop him from killing Gordon's son. He only had one hand to hold on and one to grab the boy.