Place a rack in the center of the oven and preheat oven to 350F degrees. Lightly mist a 13 by 9-inch baking pan with vegetable oil spray. Set the pan aside.
Place the cake mix, butter, milk, eggs, and vanilla in a large mixing bowl. Blend with an electric mixer on low speed for 1 minute. Stop the machine and scrape down the sides of the bowl with a rubber spatula. Increase the mixer speed to medium and beat 2 minutes more, scraping the sides down again if needed. The batter should look thick and well blended. Pour the batter into the prepared pan, smoothing it out with the rubber spatula. Place the pan in the oven.
Bake the cake until it is golden brown and springs back when lightly pressed with your fingers. 30 to 33 minutes. Remove the pan from the oven and place it on a wire rack to cool.
Meanwhile, prepare the pudding. Place the pudding mix and milk in a medium saucepan. Fold in the butter and vanilla. Cook over low heat, stirring, until the butter melts, 3 to 4 minutes. Remove the pan from the heat and let it cool for about 5 minutes. Fold in the banana slices.
To assemble the cake, spoon the pudding and banana mixture on top of the cooled cake, spreading the mixture out to all edges of the cake with the rubber spatula. Cover the pudding with the whipped topping, spreading it out to the edges of the cake with the rubber spatula. Scatter the crushed vanilla wafer cookies over the top. Serve at room temperature.