Yoga is only yoga. Without any tradition’s name in front or behind. People have to understand the essence, the real meaning of yoga which is not any specific style and not only physical exercise.
How much would Patanjali, the author of the Yoga Sutras 3000 years ago, charge you for becoming his franchisee and using his name and Sutras?
Karma Yoga is when you do something and then forget it. Whatever you do, just offer it to god and don't keep any expectation of getting something in return. That is Karma Yoga.
We should not say doing Yoga, but being in Yoga. When your soul wants to connect and unite with its source.
Hatha Yoga is food for the body, Gyan Yoga is food for the mind and Bhakti Yoga is food for the soul. Like this you get the full package.