I understand you are not using the fully qualified domain name in your deployment, but for what its worth here are the differences in my user connection:
https://website.com/arcgiswebadaptor/services
vs my admin connection:
https://website.com/arcgiswebadaptor/admin
(the 'arcgiswebadaptor' is the actual name of our web adaptor [lame I know]).
My first questions would be are: 1) are you using an account with admin privileges to make the admin connection?
2)It doesn't look like you are using the web adaptor, but have you looked into this?:
3)are you using a 10.1 desktop to connect to your 10.1 server?
If your server administrator has configured the Web Adaptor to block administrative access to the server, you will not be able to connect to the server in ArcGIS for Desktop through the Web Adaptor URL using an administrative or publisher connection. Alternatively, you can connect directly to the GIS server by using the URL format http://<GIS server>:6080/arcgis.
No, I do not use the web adapter, so I won't meet the limit access problem.
I read the post you mentioned and send a pm to Ted Cronin, hope he can give some help.
The error code is very strange, it seems that the code can be divided into two part, first: ERROR: code:404, Could not find resource or operation 'publicKey' on the system. second: Error: ..., No resource could be found at that address. I wonder if it could be caused by missing compoent of the operation system or some system security settings.