In my case it looks like it was some sort of conflict between HTML Editor and my server. When tech support upgraded the server a couple of weeks after posting this thread, the problem ceased. Unfortunately I don't know the details of the upgrade. But you could buy some webspace from another provider (free with a domain name perhaps?) and then test it on a different server to see if you still get the same problem.
DMX Zone just couldn't replicate the problem on their own servers so could never figure out what the problem was. Just glad it sorted with a server OS upgrade