Flip, you need to be careful about this. I have seen loan officers tell people not to worry about it countless times when they had no idea WTF they were talking about. Keep in mind that the loan officer needs to make "a sale" and they are like any other salesman, they will say what they think they have to in order to make it.
You watch your own back. If your payment is reported late on the 30th day, then make dam sure you have made your payment by that time.
Keep in mind that there is a 3 day recission period on refinances. That means that if you close your loan on the 29th, there is no way in hell that the payoff is going to be applied to your old loan by the 30th.