Proposals, like prayers, should be said in private.
As much as I've always agreed with that (and how it puts unfair pressure on the proposee as you and TRVA said), I honestly can say I would love something like the first video. From it alone, you can tell that everyone was confident she would say yes, and you can seriously tell how much they love each other. Putting that much effort into something that he knew she would love also shows that he knows her taste and thought she deserved something that was right up her alley.
I myself don't really like having a lot of attention on me and would be extremely uncomfortable if someone did this and I wasn't sure about things. But if it was someone I knew for sure that I wanted to be with? Heck yeah. I would love the kind of relationship they seem to have. They really get each other and have lots of fun.

I agree- it really depends on the person in question as well. If you are doing the proposing, you should know well enough what the person receiving the proposal would prefer.
Nick mentioned a few different ways that he thought about proposing to me but I think the one he picked was perfect. I was sitting at home on a Tuesday night, cutting coupons (naturally) and randomly out of nowhere I hear these silly little beeps and boops. A minute later, I see Wall-E coming around the corner into the bedroom with a ring box on top of his head. 
...
I didn't cry until he told me that he asked my Dad's permission....
Okay, that's friggin' adorable. >3<
All that being said, I thought the video was very touching, and I don't have a significant other to dream for such things from, so I'm probably full of crap.