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See, you're still not understanding the terms of your contract. You failed to make a payment BEFORE the due date. When you were sent a bill it contained a payment amount and a due date and you didn't make it. The fact that at some point in time BEFORE that bill was even generated you made a different payment is irrelevant, it has nothing to do with the bill you received and which you failed to pay. As I said it may seem strange to you, but logically it's perfectly sound.
There is a causality gap in your logic, you are attaching a payment you made to a bill you received after that payment was made, and they have to be time-ordered in the reversed manner. You probably expect that once you make payment you make should have an effect on subsequent billings and that's very likely still true but not to the extent you expect - may be if you hadn't made the payment you weren't required to make, the bill would've been for more than $15.
As far as logic and reason goes, their policies seem perfectly reasonable to me - in order to not alienate customers they forgive the first mistake, while still sending a rather strong warning. You didn't have to escalate the situation and find a person with the authority to override the penalty due to your specific circumstances, it was automatically dropped for you.
It just seems that your frustrations came primarily from your misunderstanding of your obligations, rather than from some inherently odd terms of the contract.
The failure on my part was not understanding the May 5 date. I only looked at the "due date" not the statement date. There was no previous bill to this, and May was my first statement, so as far as I knew, I made a payment before may 28th. Like I said, this was a result of this being the first bill. Later billing will be more straightforward.