The sound in the video is coming from the loose heatshields on the top of your exhaust manifolds. VERY VERY common problem with E46s.
Easy fix, get your pliers and rip them off. Its a 2 piece design , all you;re diong is ripping the top part off.
Other noises that i heard were off of a shot water pump bearing and a bad DISA, but those noises could\ve been deceived by the noise of teh loose heatshield.
I can bet $1000 thats its teh heatshield in that video, what you have I have no clue because I'd need a vid of your car to tell.
BTW, aip pump never sounds like that, its loud at times but it never rattles. Yes, it works only at cold starts.