I don't think the types of dreams you have that are unusual. I have "environments" and "life histories" that continue in different dreams and I remember having dreamed while in a dream.
I think lucid dreaming can really be seen as a continuum; there is no sharp line between lucid dreaming and non-lucid dreaming.
You can have lucid dreaming where you are aware that you are dreaming and you can control everything in the dream; lucid dreaming where you are aware that you are dreaming but still feel that the events in the dream or out of your control; or lucid dreaming where you are aware that you are dreaming and you think very logically about some things in the dream, but at the same time in the dream you accept things that are totally irrational and nonsensical and you forget things that would be very obvious if you are awake. I guess you could also have a dream where you start to have a suspicion that you might be dreaming but wake up before you find out.
Those were only examples that I thought of off the top of my head; I'm sure there are lots of other ways you could experience some aspect of lucid dreaming.
I think the fact that you kind of remember your dream having been a dream puts you somewhere in the continuum.
In lucid dreaming,
parts of your brain that would be active when you were awake stay active. I suppose this affects different lucid dreamers in different ways.