global $current_user not current user

Before answering my own question, I would like to preface it with a very well-deserved thank you for all of your comments and suggestions. It led me to the root of the issue. @toscho, it was your comment that made me think about disabling the other plugins that were activated on the site. That narrowed it down to my plugin (which I thought was the culprit) and after multiple tests (on various servers besides my host), found out that the issue disappeared. Below was the solution (which was also suggested by @toscho):

I contacted my host (SmarterASP.Net – great group of guys by the way), and explained the situation, and they said that there was an output-caching option enabled on my site. They then proceeded to inform me of how to disable it and voila! Ghost-login issue has disappeared.