The Park Tool technique is clever but unnecessarily complex.
Another way to retighten the "Rivnut" (the official name of the fitting) is to get a long M5x.8 bolt and a matching nut. Thread the nut far up the bolt and then thread the bolt all the way into the water bottle fitting. Hold the bolt head stationary with the proper allen wrench and tighten the nut firmly against the top of the fitting with an open end wrench. If you tighten the nut enough, you will expand the Rivnut and firmly fix it back in place.