Yikes--we do have at least two meanings for "admin" in the situations you're asking about.
First, and the one I was referring to with respect to the groups question is your ArcGIS Online account in your organization is assigned to have the administrator role.
Second, is when creating the ArcGIS Server Admin connection in ArcCatalog, the credentials that are provided are for the server administrator (a user role defined on the server). However, when creating a User connection, the credentials are for an ArcGIS online account, where we are internally checking to see if it is an account in an organization, a public account, or if no credentials are supplied, it is considered anonymous access.