From the Instructor:
Qigong is a very broad category - just like exercise or prayer. In essence, it refers just to an exercise that coordinates the movement of the body, breathing and one's mind (mindfulness about the movement of the body).
When the 3 cohere, that's Qigong.
Based on this definition, Tai Chi is Qigong. When created it was practised in such a way that it becomes martial (but no different from Karate and etc). Most present day Tai Chi, however, has been reframed and exercises and have no martial ability. But it still has health benefits.
With regards to Chi, it is most likely a metaphorical language to describe the experience of an increased blood flow in one's body.
Chinese exercises and meditation, as in all forms of ancient healing, often couple together simple exercises, breathing exercises with deep meditative exercises that borders on spiritism. In reality these diff elements can be decoupled. Most Tai chi teachers, if they teach in a non-religious context, would practice this. If they start to introduce forms that sound like spiritual healing, that's when we refrain. Mostly it doesn't happen.