If I run the updater again, it downloads the same update again - should it not just realise GSX is up to date and not dowwnload anything?
That's how supposed to work but, in THIS case, it's not downloading the file because you had an old version, it was downloading it again because it didn't get the latest version of the file which is used to prevent unnecessary downloads for *THAT* file. It's not a "general" Live Update problem, it was a specific problem with *that* file.
As I've said, the only thing affected here, in THIS case, was our bandwidth bill ( unless you are on a metered connection, so your bandwidth bill would be affected too ). Of course, following your report, the issue has been fixed now.