The emphasis was on both the "re-visioning" of a dream, as FryingMan mentioned (remembering, and really feeling that you are back in that scene) and also the affirmation (not mantra) that you will recognize the dream state when that particular incident comes to pass. You are in essence visualizing yourself becoming lucid, and then coupling that visualization with a strong verbal commitment. You only need to repeat this process until you feel that your intention is set.

I've heard that it doesn't work if you say Lucid Dream in your mantra
Definitely a myth. Your intention to recognize the dream state (and your ability to carry that into sleep) is all that matters.