I had the good fortune of learning yoga from a teacher whose core focus was on the breath. He taught this technique and other breathing techniques first and foremost before any movements ever came into play. The difference in the relaxation, stress-relief, etc., between that and the way I see yoga taught at these chain studios is incredible. I wish that was the standard, but obviously much fewer people have the patience to learn this way.
It's less that there are two types than it is that they're complementary aspects of a larger thing. You can avail yourself of one — to great (if incomplete) benefit, even — without availing yourself of any of the others.
The way Yoga is practiced in the US and most of the world is just a set of exercises. Without the spiritual & mindfulness it is not Yoga as it is meant to be. Yoga is designed to be the complete package, not just for your body, but also for your mind, your thoughtfulness & soul (atma).
