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When a private bank makes a loan, the dollars are not created out of thin air, no. They loan out depositor's cash.

That is not how things work in the US with banks and loans, they (banks) do actually create "money out of thin air" when they make loans:

Read the section under the heading: "How Banks Make Loans in the Real World".

creating money in the sense of lending on margin is very different from the sense in which the Fed creates money.