Welcome
Yeaaa congratz! not bad, flying in your first lucid dream. I managed to fly in my second LD.
I have had false awakenings as well and it's definitely perfectly normal (what does 'normal' even mean? I don't know )
Anyway... Sometimes I have those strange dreams that seem to be dreams within dreams (@Scionox: thanks, I never knew that those dreams within dreams don't actually exist. So they are just a different level of focus within your dream that makes you think that you are having another dream, right?).
So yea just so that you are prepared: maybe you will encounter those dreams as well where you are dreaming and then within this dream you think that you have a lucid dream. As I just learned from Scionox that's just a focus shift within your dream. This can be really confusing and quite annoying if you don't realise that you are still dreaming.

Good luck and I wish you many many more awesome LDs to follow!
- Benni