First of all, congrats!!! Awesome to hear that you made it lucid intentionally, and so quickly!
Regardless of it being a false awakening or not, this would be a DILD- a Dream Induced Lucid Dream or Dream Initiated Lucid Dream, meaning that the dream had already begun before you were conscious and lucid, and that (most likely) the dream itself is what made you lucid.
The trouble with DILD is that you can't really reliably narrow it down to "I had a DILD through the MILD technique" or "I used awareness to have a DILD". The DILD techniques are all really related to one another and parts of one are integral to parts for another (For example, if you use prospective memory for the
MILD technique, you probably tell yourself that you'll
reality check to your
dream signs and that you'll be
aware enough to notice them). An exception is when you use a device to EILD, which beeps or flashes or something, and even though you could always simply dream that the device had gone off.
But, what matters is: you had a DILD

So, congrats!
You should try to make a habit of reality checking whenever you wake up. This will help clear up whether or not it was a FA or a real awakening in the future.
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