Stimpy is correct - the. credit is available once the annual fee is paid and is for any travel booked through AMEX online or via the travel service.

The first notice was buried at the end of my June bill and they sent me a few notices when my renewal came up last month. Full terms from the June notice:

Centurion American Express Travel Credit
When you book travel through American Express Travel online or with Centurion Membership Services, you can earn up to $2000 in statement credits for eligible travel purchases made with your eligible Centurion Card through 12/31/2021.
Offer Terms
For eligible Centurion Cards that renew April 2020 through July 2020, this offer will be available starting August 1, 2020. For eligible Centurion Cards that renew August 2020 through March 2021, this offer will be available starting on the first day of your Card renewal month.
When you book travel through American Express Travel online at amextravel.com or by calling the number on the back of your Centurion Card, you can earn up to $2000 in statement credits for eligible travel purchases made with your eligible Centurion Card. You must have a booking with American Express Travel, made on or after the offer start date with your eligible Centurion Card, for your purchase(s) to be eligible for the credit. Statement credit(s) may not be received or may be reversed if the eligible purchase(s) or booking(s) are returned or cancelled. Both booking(s) and charge(s) for eligible travel must be made by 12/31/2021 to receive the credit(s). Eligible travel purchases include purchases of airfare, hotel, flight + hotel packages, cruise reservations, car rentals, and tours made with your eligible Centurion Card, minus returns and other credits. Eligible travel purchases do NOT include purchases of timeshare properties, group reservations or events, ticketing services, cancellation or other fees, interest charges, purchases of travelers checks, purchases or reloading of prepaid cards, or purchases of other cash equivalents. Purchases by Additional Card Members are not eligible for statement credits. Credits or rewards you may receive with your Card on other purchase categories or in connection with promotions or offers from American Express may not be combined with this offer.
Please allow up to 90 days after an eligible purchase is posted to your Card for statement credit(s) to be posted to the Account. Call the number on the back of your Card if statement credit(s) have not posted by that time. If American Express does not receive information that identifies your transaction as eligible for the offer, you will not receive the statement credit(s). To be eligible for this offer, you must have a US consumer Centurion Card that was solicited for this offer. To continue to be eligible for the offer, you must maintain the Centurion Card that was solicited for this offer for the full offer period, or until you have earned and received the full amount of the offer credit if sooner. To receive the credit(s) your Centurion Card must be not canceled and not past due at the time of statement credit(s) fulfillment. If a charge for an eligible purchase is included in a Pay Over Time feature balance on your Card Account, the statement credit associated with that charge may not be applied to that Pay Over Time feature balance. Instead, the statement credit may be applied to the Pay In Full bala