Hmm…so I just tested this. If I have enable names, display name on posts, and hide user profiles from public all checked, full names do appear on posts for anonymous users. I couldn’t find a setting combination that would display full names on posts on for logged-in users.
Hide User Profiles from Public is doing what it’s intended to, it’s preventing access to the user’s profiles. A “full name” is not generally a secret, nor is it part of the “user profile”, hence why it is not hidden by this setting.
What you can do is hide the names via CSS. The following CSS hides the name field on badge pages for anonymous users only.
.anon .show-badge .name {
display: none;
}