Is muay Thai and muay thai kickboxing the same thing?

Yes, its the same thing. Muay Thai is Thai Kickboxing. The biggest difference between western style and Muay Thai is that in Muay Thai, they won't break the fighters up if they clinch up. Its a very aggressive style and requires a great amount of flexibility. And yes the way of eight limbs is hands, elbows, knees and feet. You'll be using them all.