First video ....yuk
2nd video, I think a lifter issue.
As suggested above,remove the serpentine belt and run the engine, this will eliminate all the pulleys and accessories on the front of the engine as a potential source of the noise.
Now, questions?
1 How long since last oil change?
2 What type of oil are you running?
Also, as suggested earlier, use a long screw driver and some earmuffs like a stethoscope to try and narrow down where the noise is coming from.
Seeing as the noise went away once it warmed up I'm not thinking big end, if anything that should get worse as the thinner oil would make it worse.
As suggested above, grab a bottle of Nulon Lifter Free and try that. It acts like a oil flush to remove carbon build up but doesn't thin out the oil so you can drive the car and leave it in till the next oil change. If the problem is crap stuck in a lifter than the Lifter Free should help. If it's mechanical wear and tear than it won't do ****.