The language from the Delta website comes straight from the CDC website. But what continues to be a mystery is what standards the health care provider applies to determine whether the patient qualifies for a certificate of recovery.
I have seen many non-governmental websites that say that you can get a recovery certificate 10 days after your positive test result, but I can’t find that language or any other medical requirement from the CDC. All that the CDC website tells me is that the certificate must be from a licensed health care provider (and on official letterhead), and that it must contain my name and date of birth as shown on the passport.
I know that you do not need a negative test result to return to the US, otherwise there would be no need for a certificate of recovery alternative. My admittedly amateur understanding is that many people continue to test positive, maybe even for weeks and months, even though they are no longer contagious.
Anyone who gets a negative test result after testing positive for covid can just use that test result to travel, no need for a certificate of recovery, at least that’s how it seems to me.