Thomas,
Did you run the "enable geodatabase" part of the ArcSDE for Microsoft SQL Server Express Personal setup wizard? You don't need ArcGIS Server for that.
As you indicated, the database was not actually created. There's a known refresh issue in the Catalog tree - the geodatabase shows up even though it isn't there. If you close and reopen the client, the geodatabase is no longer under the Database Servers.
The usual reason for the geodatabase not being created is the SQL Server Express instance not being properly enabled to store geodatabases. In the case of the forum post you included, that was for the workgroup variety. I've not heard of this problem occurring when you run the ArcSDE for Microsoft SQL Server Express Personal setup wizard.
Try running that portion of the setup wizard again. If you still can't create a geodatabase - truly create one, not just a phantom ;), please contact tech support, as we need to log that as a new incident.