What version are these videos that are watching in?
If I'm not mistaken, the view was moved down a lot in recent versions. No one really seems to appreciate this much though, I'm sure Mojang is/has or will take note of it soon. ^^
Not sure if it is the same problem he is seeing but I noticed this yesterday... Hopped into my minecart and I couldn't see anything really but the walls of the minecart:
I've been able to repeat this behavior, but only when I hop in my minecart when it is sitting at an angle like in the middle of the V in this image:
and for reference, here is what it normally looks like when I hop in the minecart when it is flat:
I've had this glitch as well, since i went onto 1.6.2, and as you point out, it does only seem to happen if you enter a minecart on a slope. Most of the time you stay like that til you exit the cart, but just occasionally it does correct itself mid-ride. I noticed if you do 3rd person view whilst in the cart, you legs are moving as if you were propelling the cart with your legs, this doesnt happen when sitting normally in it.