While I have seen it done with moving (or even exchanging) one stitch to the other needle, I have never seen the actual knitting done the way Cat shows in her video. I LIKE IT! The second video you posted is the way I have seen it done. But with Cat's way there is not length of yarn stretched between the two tips when you get to the end of the first row, which is usually what I end up with. Of course, it all works out fine after a couple of rows, but I definitely am going to use Cat's way next time.

In short, I don't either way is wrong or right. They both work fine, and it is strictly a matter of preference and what you are comfortable with.

Thanks for your answer, Donna. It was you who first told me about knitting on 2 circulars last fall when I asked about the London beanie. I started off using Cat's method. Then later tried the other.
Still not sure which I prefer, but am glad to know there's no right or wrong.