Buckets are notoriously leaky - you can't use airlock activity as an indicator of fermentation with them. To do that, you need a carboy or Better Bottle with a tight-sealing bung or universal stopper.
By the way, shaking and stirring are equivalent in terms of maximum oxygen saturation, and either one works fine for all but the most extreme beers.
The Fiesty(sic) Goat Brewery est. 2007 & Clusterfuggle Experimental Ales est. 2009
Primary: Saison, Berliner Weisse, sLambic I, sLambic II, IPA, 70/-
Secondary: Orange Blossom Mead, Flanders Red I
Kegged: Fat Man Porter w/ 1469, Fat Man Porter w/ 005, Centennial Falcon w/ Conan, Centennial Falcon w/ Denny's, Barrel-aged Fat Man Porter, Belgian Dark Strong, Dark Mild