Solution: Update your iPhone to the latest version first, using iTunes. Then you should be able to restore the backup.
In case you get the "The iphone could not restore because the iphone disconnected." at random times during the restore, make sure, that there is no USB hub (e.g. external screen or docking station) in between the computer and the phone. Also, a new USB cable might be the solution.
Hard to believe how incredibly unhelpful those error messages are. No wonder >3000 other people have indicated here that they had the same problem.
I don't want to know what would have happened if my old phone, which I lost, would have been on a different version than the one that I bought new and the current version.
Thanks linfaTN and everyone else here for your help!
Oh, and it seems that the "sync over WiFi" option did not work, so I lost all data since the last USB based backup.