A monetary system can have two types of media of exchange: money and money substitutes. Money is the general medium of exchange. Money substitutes are perfectly secure, on-demand, at-par redemption claims for money. Money plus money substitutes constitute the system’s money stock. In our system, for example, Federal Reserve Notes are money and checkable deposits are money substitutes.
There are three types of money: commodity, fiat, and credit. Commodity money has (roughly) the same value of the commodity from which it is made as its purchasing power as a medium of exchange. Commodity money comes into existence by people’s choice within a market and can be maintained entirely by private enterprise. Fiat money has less value of the commodity from which it is made than purchasing power as a medium of exchange. Fiat money can only come into and maintain existence from the intervention of the state. Credit money occurs when the state declares that it will make something money or a money substitute in the future and people believe the state and therefore, use the item as money today.
There are two types of money substitutes: money certificates and fiduciary media. Money certificates have a 100 percent reserve of money into which they can be redeemed. Money certificates would be produced by private enterprises as warehouse receipts (i.e., owners titles to money being warehoused) of money owned by people and stored by the private enterprises. Fiduciary media have a fractional reserve of money into which they can be redeemed. Fiduciary media come into existence through intervention of the state.
With these categories as background, one can define different monetary systems by combining the different types of money and money substitutes. For example,
Free Market: commodity money and money certificates.
Classical Gold Standard: gold coin money and fiduciary media.
Current Monetary Regime: Fiat money and fiduciary media.
Then one could add contingent conditions to tailor the description of the system to particular cases in the real world. For example,
Gold Exchange Standard: gold bullion money and fiduciary media.
And, of course, there could be more complex contingencies, as say under the National Banking System.
If I were to take a guess at classifying the systems you mention, they would be as follows.
Commodity Money: commodity money and fiduciary media.
Sovereign Fiat: fiat money and money certificates.
Debt Based Fiat: fiat money and fiduciary media.
In our current system, money itself (i.e., currency or Federal Reserve Notes) is printed by the Federal Reserve. Banks bring money substitutes (i.e., checking account balances) into existence by creating credit. The process of monetary inflation and credit expansion is driven by the financial gain to the state for printing more fiat paper money (and having credit expanded by the banks) and the indefinite profitability for banks to issue more fiduciary media by creating more credit.