I would say the first defination is the closest, although you don't necesarily have to have any control to be lucid.

The other type of dreams you are describing are just vivid dreams. If you do not realize that they are dreams then they are not considered lucid...

Control is just a plus that you have when you are lucid and it comes from the realization that you are conscious, and that there are no limits in the dream world. Kind of like realizing the matrix isn't real gives you the ability to do things you would be able to if you weren't.