I know the title sounds rethorical, but tonight I hade a very very weird dream.

Not because of the contents, which were pretty random, but because I had a lucid dream within another dream I think. Or, at least, I "woke up" from the lucid into a regular dream, if I can say.

So, I was in this place with these people I know. The content of the dream is long and dramatic, anyway I somehow manage to understand I'm dreaming. I needed to focus on the dream to run some tests (touch things, focus on details, etc) so I closed my eyes and ordered them do disappear, and they did. Good, I'm both lucid and able to control, I think.
So I go around, touch some walls, rub my hands together and simple physics test. But I feel the dream fading, so I try to spin and find myself in another place. The dream got weirder: I was still lucid but unable to control the dream. Since I was on a sort of roller coaster, I quickly woke up.

But here comes the interesting part: I "woke up" into another dream. It wasn't a false awakening: it was a normal dream. The other false awakenings I had, I woke up in my bed, got up, made it to the table and start writing a dream before I realized I actually wasn't doing anything.
This time, I was at my dad's place (my parents are divorced), and the dream showed all the defects of a regular dream: illogical, blurry, space and time behaving strangely. It took me a while to realize I was dreaming again and actually wake up.

So long story short:
can I have a lucid dream "inside" a regular dream? Does it still count?
Or did I have a lucid dream and the regular dream counts as a false awakening?