I just saw you had your trade sorted out in arbitration. Good decision not paying because the names don't match. Common sense is always a good advisor when dealing with money. I just deleted the following but decided to leave it here for future reference:
There's no description for the cash deposit procedure.
When you pay, you keep the deposit slip and wait for the BTC to clear.
When the seller receives the money, he confirms in the client that he got it and you'll get the BTC to your local wallet. If he tries to scam and doesn't confirm, you'll have the deposit slip to prove your payment. You'd talk to the arbitrator once the trade times out. He will ask you to tear the slip apart and take a picture of it. You'll have to send the pic to the arbitrator. That's done within the Bitsquare client.
There's no way in Bitsquare to send a message your trading peer as it introduces additional risks.