When you do the double cast-on, you don't want to crank down too tightly. But you do want to pull tightly enough to tidy up your loop. Sometimes the yarn gets a little stubborn and you have to pull one side or the other with a little more force to snug up the loop evenly-- I think that's what you're seeing in Amy's video.
Personally, I find a super tight cast-on hard to knit from (like you, I knit continental). If I find that I have cast on too tightly, I usually rip it out and start again. The main thing about a good cast-on is that it's really consistent and even. It's a skill worth practicing, and it can be kind of mesmerizing and relaxing too.
Also, if you're worried about tension problems, try making a longer swatch -- the first few rows are hardest to work with, but after that the fabric stabilizes as it takes shape, and the tension might be easier to control.