This was definitely a lucid dream, and a very good one for a first lucid. Congrats!

The clues that this was a real lucid: " I am jolted with a thought "Wait, I can only fly in my dreams, I'm dreaming!" In this instant I feel very weird, like I'm in some alien place. I don't feel like I normally do inside my dreams. It's not totally uncomfortable, but for a moment or two I feel uncomfortably aware." This jolting and uncomfortable awareness, definitely would not happen if it were just a dream about being lucid.

Also, congratulations on not missing the false awakening, and doing the task of the month all in your first lucid dream.

See, in my first lucid dream ever, I became aware I was dreaming, so I tried to do a somersault in the air, but woke up from excitement in the middle of that. The end. So, you see compared to that your first lucid is awesome. Although, my first lucid was followed by a day during which my adrenaline level was higher than normal and I was just a bundle of joy all day long, so even my first lucid was pretty awesome to me.

Congratulations again.