So you don’t think the bank is creating money when it lends out the balance from savings accounts? (That is, the deposits that you do not have “the right to use at any time”.)
Banks don't lend out balances. "Banks lend out balances" is just an analogy. (It was truer under the gold standard.)
Banks create money through lending and destroy money via write-offs. Periodically (in the U.S., at the end of the day) banks count up what they owe and what they're owed and make sure the ratio is legal. If not, they borrow (from each other or from the Fed) to make up the difference.
(For monetary purposes, savings accounts are checking accounts. Banks don't segregate savings from their balance sheets. Balances in savings account can be withdrawn just as easily as those in checking accounts.)
There are different definitions of money and saving accounts are included in M2 (“money and close substitutes”) but not in M1.
What about time deposits? For short maturities they are included in M3. And what about longer term depositss? Are banks creating money when they lend those out?
You're hanging on to the "banks lend out deposits" mental model. Stop that. The Bank of England has an accessible write-up on this: https://www.bankofengland.co.uk/-/media/boe/files/quarterly-....
Loans create deposits. Whether those deposits then end up in a checking account through a payroll service or a vendor's savings account through a payment for goods and then transfer is irrelevant.
A bank making a loan is trading one asset, cash, for another asset, a loan. If that's "creating money", then doesn't that imply that a loan is "money" and cash is not "money"?
Down the line you suggested depositing money creates money because now you and the bank have a dollar. That’s not true, because the bank doesn’t have title to the dollar.
(And can deposit it at the federal reserve, if somehow you don’t think actual money is good enough to qualify as money.)
Custody isn’t title.
The deposit is their liability (obviously they can't spend it). The dollar is their asset. When you deposit the money into a bank that dollar is not yours anymore, the bank is not keeping it in custody for you.
The money lent does not go away when the check or whatever is cashed at another bank. The reserves follows the check. We're just working with more balance sheets now.
As long as the loan asset remains in the books, it earns interest which will add to bank equity.
We can keep bringing up details, but the basics remain the same. Bank lending creates money.
In fact banks also create money when they take cash deposits (and destroy money when they hand out cash).
If I deposit a $100 bill into a bank we’re increasing the money supply because I have $100 and so does the bank (until it spends it, trades it for a non-money asset or lends it out).
When a loan is taken, it is usually not taken just to sit in a bank account, it is to be spent. When that happens (by cash withdraw or transfer to a different bank), a bank must release/transfer base money, which it can get from cash deposits, incoming transfers or in exchange for some other assets.
Therefore, although banks create M1 money by lending, they are limited in practice by necessity to keep balance of base money. So it is more a distributed/emergent behavior, where each bank ability to create M1 money is limited by its market share and the rate of money production by other banks.