Maybe this will clear it up.
https://stacks.cdc.gov/view/cdc/112475
"CDC does not maintain vaccination records or determine how vaccination records are used, and CDC does not provide the white CDC-labeled COVID-19 Vaccination Record card to people. These cards are distributed to vaccination providers by state health departments." Emphasis theirs
Maybe this will clear it up.
https://stacks.cdc.gov/view/cdc/112475
"CDC does **not** maintain vaccination records or determine how vaccination records are used, and CDC does not provide the
white CDC-labeled COVID-19 Vaccination Record card to people. These cards are distributed to vaccination providers by state
health departments." Emphasis theirs
(post is archived)