Possible first proper Lucid Dream. Some questions

Please note that I copied this from my post on reddit about LDing. It is probably as much info as I remember about it so no point typing it twice! Thought I'd get a wider range of responses putting it on here too :)

So the last couple days I stumbled across r/luciddreaming on reddit and have been intrigued by it. Being in-between jobs I haven't had much else to do and the idea of trying to achieve some sort of lucidity in my dreams is something I guess to achieve whilst jobless.
I read into it yesterday and read about Reality Checks and one of which is trying to push your thumb through your opposite palm to see if it comes through the other side, so decided to focus on this for now.
Being unemployed, I tend to just lay in bed drifting in and out of sleep of a morning as I have no need to wake up at any specific time. This morning I woke up out of a dream that was fairly vivid but not lucid, after waking I thought of what I could have recognised out of that dream as off and how I could have tried to turn it into a LD.
I drifted off to sleep again, I'll try to recall it as best as I can. I was walking up a street with a steep hill that is close to my house, besides me there was a guy I used to work with and a professor type guy. The professor asked us (to the best of my memory) a very complex type math/physics questions. Cant recall what it was, probably just a filler that was made up I guess. What felt like a couple seconds after I sort of thought of this as being strange, like why would I be walking with this guy I used to work with an be asked this problem? So I questioned if it were a dream, tried the thumb through the palm thing, to which the shape of my thumb came out the other side of my palm, kind of like pushing it through putty or something similar. So I realised it was a dream. At this point I started feeling excited and felt like I was coming out of the dream (some people had mentioned this was a problem for the first couple), so to remedy I tried spinning which I also read yesterday as a way to stay in. I think I regained a bit of control, and ended up shooting 3 fireballs in the direction of a house at the top of that hill once I was "aware". Then I came out of the dream.
I've been mostly puzzled all day as to whether it was actually a lucid dream, or a dream where I thought I was lucid dreaming, if you can work the difference out. All the things I experienced were from things I read about yesterday, so don't know if it was a "mock" LD or whatever. So want some advice :)
Also if it helps to put in, when I was younger I remember having some nightmares where I knew just to force my eyes open to wake up or one where I realised it was a dream and woke up pretty much straight away.
Sorry for the mass amount of text, but thought it was important to be as concise as possible to get as much proper feedback as possible.

Hey and welcome to DV!

For it to count as a lucid dream, all you need to do is to be aware of the fact that you're dreaming. There are different levels of awareness though but as long as you know you're dreaming, it's a lucid dream. It sounds very much like a lucid dream to me, so congratulations!

Hello lucidintraining, I agree with Matte, sounds like a LD to me. :) Congrats, and welcome to the forum! It's impressive that you were able to stabilize your first LD. Hopefully you can make your next LD last even longer!

Yep :) As long as you knew you were dreaming, it was a lucid dream. Expectations do play a large role in dreams, even lucid dreams. This is why it is good to come up with a plan, or even rehearse what you would like to do next time you are lucid.