Credit Limit or IOU?

When you "set a credit limit" for someone, is that effectively like
giving them that much money to work with? Or does it require further
approval from yourself to get it in the form of IOUs? I'm not quite sure
how it works but I think I've narrowed it down to two possibilities:

1) The credit limit is transferred, signed, to the designated creditor.
They can then split it up (?) however they feel necessary and each
person they give that piece of credit to will now have credit with you
for that amount. Of course only people who trust you to pay that money
will accept such credit.

2) The credit limit is a local setting that's only used to keep your
computer from loaning too much money to people. The actual money is in
the form of IOUs, where the creditor sends you money (or valuable
goods), and you send them an IOU, signed by you, for equivalent value.
The creditor can then split up that IOU (?) and each person they give
that smaller IOU to, you will now owe money to. Of course only people
who trust you to pay that money back will accept such an IOU.

The second one is what I thought initially, but it does require physical
exchange of wealth at the initial stage, as well as at the final stage,
so that might be too difficult to negotiate. But that's how banks do it:
you give them real money initially, and they give you IOUs, or checques
that you can then cash. Of course they promise an infinite amount of
money on the cheques, then conveniently fine you as punishment when it
gets overdrawn, but that's another story. The USA government worked this
way too, when they collected all the gold you own, and exchanged it for
"equivalent value" in fiat money or IOUs. Of course they stole your gold
and threw you in jail if you wanted to keep your gold and didn't trust