First:
No, HatsuneMiku01, your account of the dream seems to indicate that you possessed a small amount of waking awareness, which would mean that you were indeed lucid -- or rather lucidly reacting, if there's a difference -- in your dream. It wasn't a False Lucid.
Now:
Though I hate disagreeing with not one but
two DV Dream Guides at once, I believe False Lucids are quite real and I might happen more often than people like to think.
For Gab: A False Lucid is the result of your unconscious mind, your dreaming mind, fulfilling your expectations by providing you with a regular non-lucid dream that carries in it all the things you've been hoping will happen to you in a LD, only with a complete lack of waking-life self-awareness or memory. In other words, if you spend your days hoping and believing that you will be lucid, there is an excellent chance that your dreaming mind will oblige and provide you with the feeling of lucidity, per your expectations -- and it will do so without the help of your waking-life self-awareness.
This has happened to me many, many times. I've had dreams where I was in full "control," flying, and doing all sorts lucid stuff, all without even a smidgen of waking awareness present. I've learned to look back at my dreams after the fact and really determine whether my self-awareness was present at any level in the dream. It isn't hard to do, with practice -- just a matter of thinking about how I recall the dream, how I was perceiving things during it, what things surprised me, and how involved "I" was involved in the dream. By involvement I don't mean control -- you can be fully lucid and have no control -- I simply mean awareness that this is not "me" doing all this stuff, but DC body. False Lucids are also another important reason to do RC's, as they will handily confirm during the dream if your mind is trying to help out too much.
I could go on for a while about this, but I won't. If you're curious, browse my
Treatise on Proof thread, where I think I went to some length discussing (and, yes, defending) False Lucids.
I hope that made sense, and doesn't ruffle too many authoritative feathers. ;)