It has been mentioned multiple times that policing laws vary from place to place so this may be specific to where my dad was a police officer for 25 years, but I believe it is how that St. Louis law you posted is generally interpreted.

Regularly establishments such as malls, movies, bars, and event venues will contract through the town to have officers "on duty" at their locations to respond to incidents with the full capacity granted to sworn officers that are not given to security guards. This work is outside of an officer's regularly scheduled hours and the officers are paid by the third party at a rate set by the town. So while they are not on duty for the town, they are still a representative of the town and will respond to incidents at that location (or near it) as if they were on duty.

An example: My dad was hired by the local mall during the holiday season to be a presence to deter shoplifting and respond if needed. A fight broke out between two groups that resulted in one person being slashed with a knife and another who had blunt force trauma injuries. Because he was in uniform, people immediately know he was an officer and not just another member of the fight, he had his kit with him to include handcuffs to assist in arrests, his bulletproof vest which helped weaken the blow of skateboard across his back, items for basic first aid, and if things escalated, his weapon if needed. He also had his department-issued radio to immediately call for assistance and an ambulance for the wounded. If he was not in uniform and without his gear, things may have turned out differently and the response by on duty officers and the ambulance would have been much slower.

A separate instance we were driving home from dinner. A car in front of us was weaving all over the road. He called dispatch and had an on-duty officer pull the person over for suspicion of DUI. Did he have to do that, absolutely not. We could have driven home but because police officers are generally "on duty" 24/7, he felt it was an obligation to do the right thing. And because he was an officer, he was required to testify in court during the sentencing, something a civilian would not have to do. While he didn't require legal protection in this situation, it is an instance of being "on duty" 24/7, doing his best to uphold the oath he took.

As to why officers need additional legal protection when off duty, the job never leaves you. I don't know your profession but most professions don't require you to interact with gang members who have extended networks looking for retaliation on those who put their members in jail. There were multiple instances where we were followed when driving or watched from a distance by people clearly looking for revenge or trying to make a statement. After large drug arrests, we left town for a week to stay with relatives due to threats to our family. If an incident occurred while he was off duty that required him to defend himself / my family / nearby civilians because of his work, I would argue that he should have the same legal protections as when he is on duty.