帖子上的管理员徽章和个人资料上的版主组

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2

Item 1, is intentional, so is item 2, so neither of these seem to be bugs.

For handling item 1, you did it right, you are supposed to grant Moderator status too to the Admin for the shield to show up.

For item 2, you can hide it using CSS.

a.group-link[href*=moderators] {
  display: none;
}

Just realize if there are multiple groups assigned to the user, it may be awkward.