I’ve been with AMEX for a very long time. When I got my first card, I called and asked, point-blank: What’s the actual due date?
The true answer is this (only applicable to charge cards — I know nothing about their credit cards):
The ‘Please Pay By Date’ is a suggestion. It has (largely) no meaning. They include it because many (read: most) of their older cardmembers pay by check in the mail, and it can take a while for the mail to arrive. The check being mailed ‘on time’ means nothing to them: They must receive it on time. (Although even this can be flexible at times. YMMV.)
The ‘Close Date’ is the true due date: If you do not pay your ‘Payment Due’ balance in full by that due date (and you have not made other arrangements with them), you are technically in default. You do not want to default with AMEX.
So what does it mean to pay ‘by’ the Close Date? AMEX processes online payments by 8PM MST. So long as you submit an electronic payment via the website/iPhone app/whatever before that cutoff, your payment is on time for that statement.
Other fun facts (I have a *lot* of AMEX shenanigans experience — feel free to DM if you have specific questions about edge cases): When stuff like bank holidays or weekends is in the mix, and you’re moving money around, so long as you have submitted the payment *before* the calendar date and cutoff time, even if the money is not yet in the account, so long as it is in the account by the first business morning after the weekend/holiday, they won’t know the difference.
Also, if they attempt to debit your bank account when a bank holiday is in the mix, and the bank tells them there is insufficient funds, they will silently try it again before coming back to you and asking you where your money is (and charging you a return fee, on top of whatever your bank may/many not decide to slap you with). They assume because of bank holidays both you and/or your bank will screw up something, and they don’t hold it against you.
As with all things bank-related, YMMV, but with enough time and experimentation and phone calling, you can generally come to understand the exact nuances of what does and doesn’t work, and what you can get away with (from your perspective) and still be 100% okay from their perspective.