Unprinted is a Status for "you are going to print, send, both, or neither."

Past Due is controlled by the Due Date field. Changing the Customer Terms doesn't change existing invoices. You have to change them manually.

"As soon as an invoice is created and saved the past due stamp appears."

Then you need to Customize your template and you need to Reveal the Due Date on your screen, if you want to manually control it, instead of allowing Terms to control it. You cannot control what you do not see.