Maybe we should "bump" the thread by simply posting BTT (back to top) thus avoiding any more superfluous controversy or tense exchanges!
When you register your DNA kit you are asked to link it to a specific person (presumably you) in your tree. Otherwise, it has no way of knowing who you are, especially for people with multiple trees. Note that it can't depend on the "Home" person since some people make that someone other than themselves for whatever reason. If they don't link to a person and you click on the match, you will see a drop-down of their trees and you can select one, but you still don't know who they are so you might be, no I won't say it even if it is late, "researching" up the wrong tree.
"Administered" is when the person taking the DNA test is not a member of ancestry.com, and their test is registered under a different member. The DNA match is with the administered person who shows with only initials, not the administrator who shows as a full user name.
The bulk of these are going to be people having their parents (or other relatives) tested. I had both my parents tested under my account using the "Order Another Test" button. They will show up with their initials followed by "Administered by McCorkleAustin".