No, because the money is still private, a corpo have to maintain the ledger, but certainly it's more connected to the short term treasury money market than the credit from commercial banks.
A cbdc is a new kind of money, tether is an example of free banking issuing private notes with proof of reserves.
How do you explain Tethers connection to Satellogic, a satellite surveillance company?
Yes, it's not really private, as all bank money is in the 2nd layer, which is not private, only if you are wealthy and your status allows it.
Still, the ledger is updated in the second layers of the banking system, as central banks only "print" money in the form of reserves and cash. A CBDC updates directly from the CBDC user to the central bank's central ledger.
