Some of the examples were meant to show the overdrive capabilities of the pedal. Please note on the distorted clips the drive level is turned up past noon and the level is turned down. There is no distortion being created by the recording device.
Maybe it's youtube but it really really sounds to me like there's digital clipping. I record with the BDDI all the time so I'm pretty sure I can trust my ears. There's some funky stuff happening in there. I have noticed that youtube gets very crunchy when it compresses the audio so that's probably what it is. You just might want to double check... I'm often wrong though.
And I should add that I use my BDDI on every gig I play to add a touch of overdrive. I makes my Mark Bass rig a little sweeter sounding. With my Ampeg it give it a little more crunch, and for my GK it helps take out some harshness.