No, you do not need a separate registration.
https:// are just protocol specifiers. The name will resolve to the same host. However, once you're on the hosted machine, the web server determines what to display. Often, HTTP (port 80) and HTTPS (port 443) can be routed to different pages. This should be configurable by the website admin/tools.
In some cases, you might need to purchase an SSL certificate to match your domain name. This provides a chain of trust, verifying that your site is who it says it is. If you haven't done this, you may be using the host's "default" ssl certificate, which can throw name mismatch warnings at you (which might be why you thought it was a compromising website).