Fa and Ld

Does a false awakening have to be the event of waking up in your own room and doing morning tasks or can it also be the feeling of waking up no matter where?

Because what happened was that I was having a dream and I did a reality check, I thought I had woken up to find I was in the exact same dream, so I did the same RC and I woke up again, but this time I realized I was in a dream so I became lucid, and while in it, I remembered waking up several times before and for some reason I thought I had died and was trapped in the some crazy false awakening circle, so I tried to force myself out of it...

no you don't necessarily have to be in your room or doing your normal morning tasks... just so long as you believe you are actually waking up and not sleeping anymore. frequently I'll FA in the bedroom of my old house or something, realize that I don't live there anymore and begin LD.