You were right. My HKLM/Software key had full control for Everyone, and no other users specified. I went to another XP machine, inspected the
settings there and copied them over. That fixed the problem.
Must have been the virus I had recently (the only one I got in the
last two years, without running any AV software).

Btw, it was sufficient to install as a member of the Administrators
group, I did not have to use the exact Administrator account.

Thanks again,

Karl