Here is the solution I stumbled upon while playing around with the code. First, I removed the -domain=> entirely. Unless I specifically need that, the browser should (and seems to) use the existing hostname/domain without any problem. Second, I removed all of the header that is sent to the browser to set the cookie and replaced it simply with this: 
print $query->redirect(-url=>'http://mydomain.com/beta.cgi?c=TestCooki
+e', -cookie=>$cookie_out);

[download]
That has apparently solved my problem. I can use it successfully in MSIE 6, Opera 6, Netscape 4x and Mozilla 1x.

In reply to SOLUTION by Seumas
in thread Stupid CGI.PM cookies! by Seumas

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