In my experience out of roughly 20 police stations, only one asked for a lease contract, etc, and I had to call 12345 to resolve it. All others did it within 5 minutes once I gave them the address, passport, and phone number, and were super nice and friendly.

The thing about locals having to be registered is that the property owner has to report to the local PSB who is staying in (and not that they have to register when they come back from overseas) and from what I understand every city has different systems (from the IT point of view) whereby in some cities the actual app is the same for foreginers and locals and in some cities there are two distinct systems. Some local apartment bosses showed me the systems and they are confusing as hell

For me as someone who cares about having no problems in the long term with the country I prefer to overprotect and just do the registration anyways and while technically you need to do it at every place where you spend >24 hours, it is the first one after the entry that is crucial.