Is rule like that really existed? Did any forum admin/moderator ever say user can have local rule on their thread? There's an option for moderating our own thread; self moderated thread.
Yes, there is self-moderation option but you can only choose it when you create your threads. If you create non-self-moderated threads, you will unable to change it to self-moderated thread later.
There are some boards you can not create self-moderated threads, Meta board for instance.
In most sections, you now have the option of marking topics self-moderated when creating them. In self-moderated topics, the OP can delete replies. The option for enabling this is under "additional options". Topics cannot be converted to self-moderated topics after creation.
There are no rules to self-moderation. In self-moderated threads, replies belong to the OP. In other threads, replies belong to the respondents individually. Think carefully about whether you want to reply to a self-moderated topic, as your post may not be given due respect.