So, from my reading Szilard's answer and Landauer's answer are slightly different. But the descriptions on the page you linked vs. the Maxwell's Demon page are slightly different, and so that may be the source of my impression. It seems that Szilard claimed that acquiring the information is where the entropy gets balanced, whereas Landauer claims that restoring the Demon to its original memory is where the entropy gets balanced. Regardless of whether or not both are correct / deserve to be called the 'information entropy explanation', Landauer's is the one that inspired my original explanation.