I don't mean to get in the weeds, but I had a different impression, so correct me if I'm wrong. To me, 'private server' was like 'private sector.' Anything not on a .gov dedicated server, controlled by a government contractor was private to me.
No, the separate domain is questionable but not as big of a deal as having the private server. This is different from public sector versus private sector. She technically could have had a .hillaryclinton.com email domain on a government server and have been secure even though the domain was private. Her having the private server meant the email domain she was using lived outside government security and messages channled through it never touched the proper security measures.
In this case, Obama may have known she had a private email address, but he would really not have a way of knowing if it was being hosted on a private server or a government secure server.
Here is an example how it would work.
You have an email address edpc@state.gov - that email domain lives on a government server.
You created another email domain edpc
@edpc.com. You could work with state.gov to have that email domain securely hosted on their servers and be in complaince even though the domain is your private one.
OR
You can do like Hillary and buy a server and host the email application yourself bypassing the State Departments security and tracking requirements.
This is why, in this particular example, we really can't say what the President knew just from here because just seeing
@edpc.com doesn't tell him where the actual email was hosted from.