Actually the "toy model" is exactly how it works. When you take a loan from the bank and deposit their check your bank must pay whoever the amount on their check. There is not some mysterious ledger they just mark.

The fact that a bank takes an illiquid asset - your promise to pay them back - and turns it into a liquid claim on themselves is not creating money out of "thin air". Your promise to pay the bank is not "thin air" as you will quickly find out if you break your promise.