There are degrees of lucidity. Sometimes you can know it's a dream, and still be scared to death of the monsters, or decide that taking off your clothes would be embarassing, or think illogically--i.e., mistaking a drem character for a sleeping person, or accepting illogical things. Sometimes you can have a little control--like being able to do telekinesis. And sometimes you're in complete control and you can change your environment at will. They're all lucid dreams, though--nothing "faux" about 'em. Being able to wake yourself up is a definite sign that you were lucid.