one thing you could try is pull the ps pump belt and see if it does it in the morning. if it does... well you know. if it doesn't... got some more to think about.
now one thing bout the video has me thinking... (I went back and have been listening again)... that noise seems to be to be a fairly mechnical, almost a gear or something grinding. Could be PS idler.
I wonder if the starter drive gear is hanging out at the flywheel and the noise starts the gear is vibrating back into the starter, or maybe ice in a coolant line getting chewed by the water pump.
can you isolate an area of the engine where the noise seems to be coming from, like maybe start it then get in front of the truck real fast (or have someone else start it) and listen?