OCI 6.4.x uses Java 6 internally. The OCI client will not work with Java 5.x.
The Java install registered with the user profile you are logged into the server as was not necessarily the one someone installed to use OCI.
The OCI desktop shortcuts on the server are hardcoded to use the Java 6 install within the OCI install path, not the
Java registered with the OS / web browser, so if you are launching the OCI client on the server successfully, all I can think is that you are doing so via the shortcut.
I don't know why Java 5 is installed on the server, but removing it should not impact OCI
Matt