BP’s own Dayn Perry did the same analysis and found that each player did very slightly better in their walk years, compared to the two previous years.

The major difference in methodologies was that Perry used “prominent” free agents, whereas the first article looked @ all free agents. The linked BP.com article sums everything up nicely:
“In the end the results are about what you’d expect intuitively. Major League Baseball is so competitive at all times that one would assume that it is rare indeed if a player can turn it on one season and turn it off the next. With the average Major League career historically spanning just over four years, players are aware of just how quickly they can be out of baseball if their performance slips. So if there are indeed players with the ‘talent flexibility’ to perform at will, they are likely few and far between.”

That last paragraph can also be applied to any anecdotal “performance increase” argument, e.g. the “clutch hitter” argument.