Welcome to the forums. Paying the balance in full by the statement closing means that there was still a period between the due date and the statement closing date where there was a balance, so that should result in an interest charge since you did not pay the full statement balance by the due date. So even if they were to apply the payment on the statement date bringing it down to zero, it's very likely that your actual balance at that point would not be zero.
While a payment may be "applied" on the date of the statement closing date, it doesn't necessarily mean that the statement would reflect that. That wording is in there more for scheduling a payment on the due date, so there would be no interest accruing and loss of grace period, for example, if the payment were made on the due date but not reflected for another day or two (which would then most likely be reflected on the statement since that is generally a few days later).
Okay, that would make everything different. 
If you paid the entire previous statement balance by the due date, then yes there is a chance that you would actually end up with a $0 balance if you're paying all of the current month's charges and you can ignore everything I said about interest and grace periods. As to whether they can actually process and record the payment fast enough to reflect on your statement generated the same day, I honestly have no idea.