What's the definition of ”Due day” ?

"Due day" usually refers to the date something is due for action. It INCLUDES the due day itself. However, the usual practice for due day in legal or financial matters is before 5.00pm of that day.
The point with your case is, credit card payments are not instantaneous whether you use an ATM or at a bank teller counter to make the payment. This is mainly due to the fact that a lot of banks have a seperate credit card information management / accounting system so to seperate the credit card transactions from the main banking system. E.g. a BOC VISA card is actually issues by BOC Credit Cards Limited and not by Bank of China. This is because banks have a definite business hour (e.g. 9.00am - 5.00pm) but credit card transactions go on 24/7.
So when you pay money to your credit card, the payment to is initially credited to an account called " suspense account" inside the bank's computer system. The money is then transferred from the suspense account to your credit card account in the credit card computer system after the cut off hour which is usually 5.00pm. So if you made your payment after 5.00pm then your payment would not be credited until the next working day and yours will be treated as late payment.