An Amex third-party card is a card that uses the American Express payment processing network but the card is not issued by American Express.
For example, Wells Fargo bank may issue a credit card and select the American Express network to process transactions between the merchant and Wells Fargo.
The card would be a Wells Fargo card but you will see the Amex logo on the card as well because Amex serves as the processing network. Amex may also provide some of the benefits that come with these third-party cards.