Chartalism is often described as the state theory of money. It explores the relationship between money and an accounting system, a governance system, and a legal system. MMT applies chartalist principles to make specific assertions about modern financial, political, and legal systems.
From what I can tell, your scenario is an example of chartalism, but it also applies to a wide variety of societies.
MMT, on the other hand, is focused on a specific historical context, it analyzes “modern” money exclusively.
You make some good points that help to clarify these issues.