Bitcoind provides a "time" and "timereceived" field for each transaction, even before the block clears... I thought.
Yes, but there's no way you can prove whether the site owner's bitcoind received the transaction today or tomorrow. The received time is potentially different for each bitcoin node.
"received time" is local to your running instance of bitcoin.
"time" is the actual original timestamp used to generate transaction in client.
Bitcoin network will ignore a new transaction if it is having too old a timestamp, or timestamp from future, when it is introduced.
BTCDice works using the UTC timestamp from transaction to determine day of transaction (in UTC) and then process using secret of that day (in UTC).