Half Lucid Dream?

I've been reading about lucid dreaming over the last few weeks, and I've started doing reality checks etc. I had one lucid dream a while ago, where I was lucid for around 20 seconds managed to fly for a bit, then promptly forgot I was in a dream. However, last night I had a dream that was...sort of half lucid.

From what I remember, the original dream had devices designed to make people have lucid dreams - and a woman there helped me go under (I know, it's very Inception-esque) into a lucid dream. So, in the dream, I fell asleep and started dreaming.

However, when I was in the second dream, I was completely aware that I was dreaming, on that level. The reality check of holding my nose and trying to breathe worked, and I also had some level of control over the events in that dream level - I couldn't fly or break physics, but I could influence what characters in the dream did.

I went into this 'dream within a dream' 3 times, and each time I had the same experience - reality checks worked, and I had some level of control. When I 'woke up', however, it was a normal dream - no control, and no idea that I was dreaming.

Is this normal? I hadn't found anything about it while I was reading.

EDIT: The thing that stuck out to me was that there was an incredibly clear divide - the levels of dream didn't bleed into each other, and I was always very clear as to where I was - I 'knew' when I was dreaming and when I was 'awake'.

That is all one single dream. So that dream was a lucid dream, even though you were only lucid half the time. That is a very interesting and cool dream.