When I have lucid dream, I know that i'm sleeping in my bed (street location, city) and which day of week it is. So I have access to real life knowledge.
So I believe there is different level of lucidity, depending on each individuals.
Yes, I used to practice lucid dreaming and parent post's description is different than what I experienced. When entering the lucid state I was similarly fully (or nearly fully) context aware.
My problem was always maintaining the lucid state. Doing anything to change the dream risked awakening early and I never got great at maintaining the state for long.
So I believe there is different level of lucidity, depending on each individuals.