i just had that problem a few minutes ago, so you aren't crazy. either that or both of us are crazy. i popped the 2004 cd out and then put it back in and it works. maybe sequence has something to do with it.
You are reading too much into a coincidence. The probability of specifically this web site having any impact on the Watchtower Library for only certain people is approximately zero.
There is however a chance that Internet Explorer could, under certain circumstances, cause problems with the Watchtower Library program (or any other program that uses DLL files associated with the web browser), though there would not need any other IE windows to be open for that to occur. The most likely scenario would be if incompatible versions of one or more DLL files have been installed by either the Watchtower Library (unlikely because not everyone has the problem), or any of various other programs.
In any case, the best thing to do in that situation would be to first try uninstalling and reinstalling the Watchtower Library software, and if that fails, try reinstalling/upgrading your version of Internet Explorer. (Whether you use a different web browser doesn't make a difference, because IE is strongly integrated into the Windows environment, and is intrinsic to many Windows DLL files, such as wininet.dll, which is linked by the Watchtower Libarary software (as are many other Windows-based programs).)