I have been told by some that bitcoin exchanges do NOT record BTC transactions by a buyer/seller on the blockchain, but rather hold them in their own name and ledger - eg, if I buy one BTC via an exchange, I do not own a UTXO on the blockchain, but simply have been 'assigned' a BTC that the exchange owns.
This of course, would give rise to counterparty risk, even ignoring fraud risk.
Truth or fiction?