Lucidity means awareness. It comes from Luce, meaning light, like Lucifer, Morning Star.

Lucid dreaming means you know you are dreaming.

I think there is confusion because of different types and levels of awareness, and also Dream Control.

You can have different levels of lucidity. You may fade in and out of lucidity. Your dream control may be tied into your lucidity but not necessarily.

If you know you are dreaming, you are lucid.