Is there something important that I'm missing?
I think the basics are missing and not just from you but the public in general which is why it' s sounds so complicated. Lets start with the four types of money issued in the United States:
- Coins - physical money created and issued by the U.S. Treasury and sold at face value to the Federal Reserve, which books coins as an asset on its balance sheet. Unlike the other three forms of money I will outline in this answer, coins are issued without any corresponding debt attached to them. Coins are legal tender, meaning that they can be used to pay, "all debts, public charges, taxes, and dues." 31 U.S.C. § 5103.
- Cash - aka, Federal Reserve Notes, is physical money issued by the Federal Reserve (though created by the Treasury's Bureau of Engraving and Printing at the Fed's instruction) and sold by the Fed at face value to commercial banks, which pay for cash using their reserve accounts at the Fed, which get debited in the amount of cash obtained. Cash is one of two liabilities on the Federal Reserve's balance sheet, and an obligation of the U.S. government. Cash is legal tender, meaning it can be used to pay, "all debts, public charges, taxes, and dues." 31 U.S.C. § 5103.
- Reserves - this is electronic money created and issued by the Federal Reserve in exchange for U.S. government bonds and bills, that is, IOUs from the U.S. government. Reserves are the second liability on the Federal Reserve's balance sheet, but reserves are assets to commercial banks and other entities that bank with the Fed, including the U.S. government and foreign central banks. Reserves, that is, Fed liabilities or IOUs, are money only for these entities, and exist only as the result of government debt. The amount of reserves in the system is controlled by the Federal Reserve, which increases reserves by buying government bonds or agency mortgage-backed securities and decreases reserves by selling government bonds or agency mortgage-backed securities. Reserves are not legal tender.
- Bank money - this electronic money created and issued by commercial banks in exchange for debt, public or private, taken by borrowers from the bank, whose electronic accounts at the bank are credited in the amount of their new loan. Bank money is a liability to commercial banks but is an asset to non-bank entities who bank at commercial banks, including people, non-financial businesses, non-bank financial businesses, and governments...bank money, that is, a commercial bank liability or IOU is money for these entities. The amount of bank money in the system is controlled by commercial banks, which increase bank money bu buying debt, crediting bank accounts in exchange for loan paper, and decrease bank money by not issuing new debt as old debt gets paid off. Bank money is not legal tender.
I think what is confusing and it was even for me, is all the discussion that we hear in the media, but none of the clarification. That discussion is on reserves. Reserves function as money for commercial banks, which have accounts at the Fed, but do not function as money for ordinary people and non-bank businesses, who do not have accounts at the Fed. For ordinary people and non-bank businesses, money isn't reserves, it's bank money. Cash is an altogether different animal for the purposes of this discussion, which pertains to digital money only. But the Fed cannot and does not create bank money. Commercial banks create bank money.
For commercial banks, reserves are asset-money, and bank money, that is, ordinary checking and savings accounts, is liability money. Makes sense so far?
So when the Fed buys assets from a commercial bank or the U.S. government, it creates new reserves and credits in the bank's or the government's Fed account in the amount of those new reserves, and the seller transfer the asset to the Fed in exchange. The key takeaway here is that this simple two-party transaction structure results in an increase in reserves, but no net increase in bank money. The reason there is no change in bank money under the Fed's normal asset purchase route is that there is no non-bank entity, and thus no bank money, involved in the transaction, which is simply a two-party transaction between the Fed and a Fed account holder, a commercial bank or the government, conducted entirely in reserves.
With all that said, the traditional two-party asset purchase structure was recently abandoned by the Fed in February of 2020, however, replaced by a three-party asset purchase structure of getting public money into private hands. Since then what we have been seeing is the lockstep increase in both reserves and bank money.