Dansk:
But soul or spirit would infer that "something" such as a "person" is passed over from life to life, which is NOT what rebirth is about. I suppose it is the "essence" of the person (English is very limited).
Essence seems like a good word to me. If English is limited then perhaps we can just borrow the word from another language. (Something like chi or katra).
I prefer the Dlaia Lama's meaning above. How does this confuse the issue?
Because consciousness means the state of being conscious i.e. being aware of one's environment and one's own existence and thoughts (or similar). Redefining it to mean some ineffable essence that is mysteriously passed on at death from one organism to another will necessarily cause confusion, especially when the organisms are for the most part not aware - conscious - of the transition.
It is exactly what Jonathan Millar said, also, i.e. "Where does consciousness go?"
It doesn't go anywhere. Where does the light go when you switch a lamp off? Consciousness appears to be a property of a living brain. When the brain (or part of it) is "switched off" the conditions necessary for consciousness are no longer there, so the phenomenon ceases.