I think my post answered this, but just to be clear:
It will still be registered here, but it's yours. The $9.95 will probably be pulled from the refund, which pays a year. Registration and hosting are two separate things. You can point the nameservers where ever you want, even before cancelling your plan here.
You can't delete the info, but there's no need to. Your options:
Registration stays at DH, you point the nameservers to any host you want.
Transfer registration to any host/registrar you want, but you'll probably have to wait 60 days to do that.
Also, note that you don't have to have it registered with any host. You could register it with someone like Godaddy, Namecheap, etc... whether you host at DH or any other host.
If you're next host offers a free domain on transfers, you'd have to wait before you could do that. That really just depends on the new host's policy, if they even offer a free domain (not all do--and some might not on transfers, but just on new registrations).
One other thing: If you transfer the registration, you don't lose the year you pay for here. Registrars will add 1 year on transfer.
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