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Discover's site doesn't mention Amazon Payment as being eligible:
https://www.discover.com/credit-cards/cashback-bonus/cashback-calendar.html
Amazon.com purchases include those made through the Amazon.com checkout, including digital downloads, Amazon.com gift cards, Amazon Fresh orders, Amazon Local Deals, Amazon Prime subscriptions and items sold by third party merchants through Amazon.com's marketplace.
I actually just did this myself the other day.
I purchased a gift card for my grandparents back in Ohio - a Cracker Barrel gift card, and it had that option. I paid with my Discover card through their "Pay with Amazon" feature. I just checked my statement, and it's showing up as "API*shop.crackerbarrel", not Amazon - so yes, it does not get coded as an Amazon purchase.