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I found it really weird when i was in the lucid state to be frankly honest like it was just a daydream but vivid kinda weird.
You would really know the difference between being lucid in a dream and having a vivid "daydream". It is possible you did fall back asleep, had the dream about chasing this person, then instead of becoming lucid (which in this case would have been a DILD), you may have, in fact, woken up. Just not entirely woken up as far as opening your eyes and getting up. So you were essentially carrying on the 'plot' of the dream in your head while lying in bed with your eyes closed in a half-asleep state.