I've just been looking at a video (by an American called Autumn Wintersgill - yes, really). She calls what the OP has described - taking the right hand off the needle to put the yarn round - the 'English method', and the 'flicking' style, where you have the yarn round the index finger and just flick it over the needle, the 'American method' .
Well, I'm sorry to disagree Autumn Wintersgill, but all the knitters I know wrap the yarn around their fingers and 'flick' it round the needle. It was the way I was taught by my very English mother, who in turn was taught by her very English mother. It was also the way we were taught at school.
Gosh, if that's how American knitters think we knit here, they must wonder how we ever get any garments finished!