Can I nurse with my baby on the same side as the shoulder strap instead of the opposite side as shown in the video?

If putting your baby on the strap side works for you, then by all means do it! That's the great thing about slings - they're so simple and versatile that you can use them any way you want. When your baby gets bigger she'll probably need her feet sticking out and laying the opposite direction like the video shows, but that will be easier when she's bigger.