...and, importantly, since you were successful having your passport saved, after you scanned your passport at a kiosk you must have then clicked YES on the kiosk prompt allowing Delta to save your passport information in your Delta profile. This is an easy step to miss (been there, done that).

To recap for the benefit of others, in order to get a passport saved in the Delta profile with the "Verified by Delta" flag,
1) check in at a kiosk for an international flight,
2) scan the passport at the kiosk, AND
3) carefully read and respond to the next kiosk prompt and give Delta permission to save your passport information in your profile.

There may also be a way to do these steps using the app but I've never successfully done that. No Delta agent has the ability to flag passport details as "Verified by Delta" - only the kiosk check-in flow (and perhaps the app) have that ability.