“Must be paid the day of” means “must be paid on that day,” correct?

"on that day" <~~yes

If it read "by the day of..." then the payment could be made any time between now and that particular day.

I would assume it must be paid before or on that day.

Yes, that is correct. When someone says "must be paid the day of," it means that the payment needs to be made on that very day. The phrase indicates a specific deadline or time frame for making the payment. To verify the meaning of any phrase, you can also consider the context in which it is used.