"the financial institutions used code 2 -Early withdrawal exception applies."

Some of the issuers have the taxpayer requesting the withdrawal sign a self-declaration disaster form. Some don't ask for this. It depends on the account and plan type, for allowed withdrawals or requested distributions, etc.

For instance, a financial institution doesn't know you qualified for coronavirus distribution, because HIPAA means they cannot ask you if you are/were sick.

"But you are saying it doesn't matter to the IRS which of the 2 codes they use?"

You still need to identify what is the exception. The code for money Out isn't the tax treatment. For example, an "early" withdrawal that the taxpayer rolls over in the required timeframe means it is no longer "early" withdrawal subject to penalty, for the tax payer. The Issuer of the 1099-R still was told to make a distribution and knows the person isn't of age.