No, it's not HCI's website. Look at the url. I've done a bit of research on this. There are three different, but associated websites where Dells company name appears. It's not only HCI that gets any cut from the purchase of a Dell. Follow this link to read the terms of these sites. Read all the terms carefully, particularly #4. Follow the links that are mentioned in #4, and see who else gets a cut. Pay close attention to how these three websites are described in #4. If anything, Dell is just covering all the bases by being involved in this program. This whole issue will become alot clearer once you've read these terms.
Thank you for some excellent investigative work. Apparently HCI is using a third-party source (EduOrg.com) to provide its affiliate connection to Dell, so Dell might not even know that HCI (like many other organizations) is acting as a subcontractor to EduOrg.com and is getting a cut from Dell's sales.
I checked out the link you provided, and Section 4 of EduOrg.com's Terms & Conditions explains that it provides a virtual shopping mall for other organizations in three categories: Education Organizations, Politically Conservative Organizations, and Politically Liberal Organizations. (Obviously HCI is listed under that third category.) Section 5 says the participating organization (like HCI) gets half of the any sales commission, and EduOrg.com gets the other half.