'Deposit-taking institution' is pretty much the original definition of a bank (see 'Banking Act 1979'). But, putting that aside...
"If your loan customer wants cash, or wants to move some of that loaned money into another institution... whatcha gonna do?"
The same thing you do every night: borrow money overnight in the interbank market. If this becomes a regular thing, liquidate some assets.
"Your nonbank credit card company would run into similar problems if they didn't have any income and simply were letting people buy products with magic printed money"
When a nonbank lender grants a loan, it cannot create money in the same way that a bank can. It can't disburse the loan without having the money on hand already. By 'on hand', I mean 'in an account at a bank'. The source of that money could be:
- (equity) investors
- interest earned from other loans
- wholesale funding (money borrowed from other lenders)
- proceeds from selling loans to other parties (directly or via securitization)
You don't have any assets in the scenario we're talking about. Nobody is going to lend you money, because you're not a bank, you're a sham scheme trying to start up a magic money-printer.
You can't disburse another institution or to cash without having some assets somewhere to back all of this up. Your balance sheet, ultimately, has to make sense. Your supposed bank here couldn't loan me money to buy a car, because the dealer isn't going to be happy with numbers on a ledger in my account at this bank, they want something harder.
I'm not aware of any country that will allow you to start a bank without any assets.
I'm sorry, I don't think any explanation I give will satisfy you, so I won't comment further.
If your loan customer wants cash, or wants to move some of that loaned money into another institution... whatcha gonna do?
Your nonbank credit card company would run into similar problems if they didn't have any income and simply were letting people buy products with magic printed money, but "credit cards create money out of thin air" doesn't get tossed around in the same way as it does for banks.
(Also, your claim about the deposit is entirely false for many loans. Every car loan I've had, for instance, has been either paper or electronic checks for delivery to the dealer, I've never had that money in my own deposit accounts. It goes straight to a deposit account at (usually) another institution, where it would damn well be noticed if the lending bank couldn't actually make good on the funds.)