Okay, so with my basic electronics knowledge...
Capacitors will not rectify a waveform in the way you're talking about. What you're showing in your first picture is a half-rectified alternating current waveform that's gone through one or two diodes (can't remember). To fully rectify the waveform to straight DC like your second picture, you need four diodes. Two of them rectify half of the waveform, the other two get the other sweep. If you're only rectifying half, you get power spikes and drops, just like with normal A/C, just at half the severity and no polarity shifting. It's all + or - voltage.
The capacitor can help in the half-rectification scenario because half the time the lamp is effectively not getting power. The capacitor should assist with smoothing out those transitions, but will not be a fully straight DC output.
Believe the "premium" LEDs are using capacitors in order to prevent ghosting. When there's not enough current to actually supply the capacitor from the bleed voltage present on the shared line, then no partial lighting of the lamp occurs.
Someone slap me down if I'm wrong. I'm still learning and I don't mind if this is incorrect.
*edit* Okay, apparently I have *no clue* what I'm talking about. i can't listen to the audio on that vid right now but I see what you're talking about. I have no idea why just adding the capacitor would change the waveform like that. Hrm.