Let me add to this...
I've also tried registering the database to ArcGIS Server via Desktop 10.1. In the Catalog tree, right click the GIS Server connection and click Server Properties. On the Data Store tab, it's basically the same window as in Server Manager. When I input the name and connection string, I'm getting the error: "The ArcgIS Server is not able to connect to the specified database. Please ensure the database client libraries have been installed and the ArcGIS Server can access the database."
This seems to indicate a problem with the Oracle client on the client machine. The client machine has the 32-bit Oracle client on it because it's also got Desktop 10.1 on it, which is a 32-bit applciation. I can successfully connect, and administer, ArcSDE on this machine....
So, my environment looks like this:
ArcSDE 10.1 64-bit on a 64-bit Linux server
ArcGIS Server 10.1 is 64-bit native, and on a 64-bit Windows server
ArcGIS desktop is on this same 64-bit machine, but runs as a 32-bit app, and therefore....
Oracle client on this Desktop and Server machine is 32-bit
Anyone?