I think the difference between RL awareness and dream awareness is that in normal dreams the conscious brain is pretty doped up and slow, so getting the conscious part to have that "ah-ha" moment is the key to lucid dreaming. I've not really cracked it yet, or I'd be getting lucid a lot more often, but often the lucid moments for me have been something unusual where it's been enough to jolt the conscious brain. Often very bizarre things go unnoticed, but slightly less bizarre things have triggered lucidity in me.
One was when the page on a manual that I was reading suddenly became animated like the paintings in harry potter.
I think the point about trying to be more aware both in RL and bleed across into your dreams is that you're then more likely to have your conscious brain just a bit more skeptical and a bit more awake than normal.