If Post Published Date or Modified Date is 1 Year or Older, Display Notice on Post Page

Right now you’re just checking for the post date.
You’ll need to check for ‘post_modified’ aswell. You can find the complete post-object here: https://developer.wordpress.org/reference/functions/get_post/#user-contributed-notes

You can modify the check to look something like this:

if (strtotime($post->post_date) < strtotime('-1 year') && strtorime($post->post_modified) < strtotime('-1 year')){
    echo 'Old Post';
} else {
    echo 'Not Old Post';
}

That should check for both.