Follow-up after temporarily enabling the tool (whose UA incidentally is a confidence-inspiring "Snoopy v1.2.3"):
They're wrong. On two domains they reported 200 responses for both with and without www, although one of the two is supposed to be only without and the other is only with. I was all set to swear at my host-- I've set it as a pref in their control panel-- but tried it by hand first. And got correctly redirected in each case. Logs confirm 301 for the "wrong" version.
? ? ?
I suggest you look at your own logs and see if they match the reported results.