The cookies in IE are a funny duck. In past years, they were stored as a physical file in that INetCache folder. They are not anymore. A physical file is actually put there, but it's just a placeholder for backwards compatibility. The actual cookie is stored in a database, that is shared with Edge. The Internet Options functionality that deletes the cache and cookies seems to delete the IE cookies from that database, as well as deleting the physical "cache" files, but it seems to take a couple of minutes to actually clear out and get that database working right. In the meantime, it can still "see" the cookie if you go back to the site.
We are getting ready to roll out a change to WebParser that I'm hoping is going to solve this.
We are adding a new option for a WebParser "parent" measure, called Flags. I won't go into that fully here, but one of the flags you can set is NoCookies, which will not allow any site to set a cookie in WebParser. Any command to set a cookie is just ignored. Time will tell, but I'm thinking that this is going to help with issues like this one.
This change will be out soon, maybe later today.