The reqular Notes client package has an option to install for “Just this user” or “For all users of the workstation.” The Admin/Design client only supports “Just this user”.
When you un-installed the old client it may have failed to completely clean up the Lotus software key in the Windows registry. Thus, when you tried to install the new Admin/Design client, it still sees a key that indicates this would be an “upgrade” of the code trying to install “single-user” code over the top of “multi-user” code.
Un-install the Notes code using Add/Remove programs. Delete any previous code directory, (defaults to C:\Program Files\IBM\Lotus\Notes”). Then completely delete the HKEY_Local_Machine\Software\Lotus key and try the install again.
Make sure that you have a good backup of your previous Notes Data folder if you need to keep anything in there.