The serial number you have is apparently one that only covers a limited set of recorders - not including your Lite On DVD recorder. The error message you report makes me fairly certain that this isn't just a case of needing to upgrade to a later version to get support for your recorder.
One way ahead is to purchase a Nero 6 upgrade using your existing serial number - that will give you the full version of Nero 6 with no limits on which supported recorders you can use. The full version includes access to the main Nero Burning ROM application, the antivirus scanner and the ability to use up to four recorders simultaneously - none of these are in the OEM version of Nero 6.
I don't quite understand the rest of your question. If you're talking about installing one copy of Nero (i.e. the same serial number) on two machines, I think you'll find that's not allowed by the licence agreement, but you'd have to check. I think Nero is licensed per computer.