Okay, no one is wrong, the code may have been updated after Tim or I last saw these parts. Here is the complete picture:
1. Every day that there is no lord of the region, gold collection is reduced by 50%.
2. On tax day, regions without a lord lose another portion of their taxes already collected to administrative costs based on their region type.
3. Thus, if the Banker manipulates the books before tax day for lordless regions, they salvage a part of the already reduced tax collection.
Of course, this applies to cities as well. I hope this clears things up.