So, yes, the scene basically plays out as follows:
In the flashback, Quell asks why she (Rei) did this, and Rei responds with "they gave me life", and a few seconds later they're destroyed. Then Kovacs pieces things together (after finding the conveniently easy to access plot-sleeves) and deduces her prior claim to not remembering those final moments but also her stack miraculously surviving the explosion are inconsistent. She had to have been backed up. He then asks her why she did it, and Rei tells him Quell asked her the same thing, and that her answer then was "because they gave me life". Rei then gives the "real" reason, saying she really did it all for him.
Nevertheless, you're right there appears to be a continuity error, as there's very little time between Rei responding to Quell and the explosion for a backup to occur in. I didn't see the telltale eye movements in those moments on a rewatch. Nevertheless she correctly asserts that Quell asked her the same question, and what her answer was.
Getting the answer right is easy. This wouldn't have been a spur of the moment decision, as the Protectorate needed time to enact their attack plan. Rei would have thought this out ahead of time. So it should be easy for her to know what her answer to that question would have been, as she would have played it out and analyzed it many times in her head before then.
Getting the question right is a little tougher. I'll give two possibilities, one of which is based on a subsequent episode and I'll put in spoiler tags.
First possibility: she spent enough time with Quell to understand her well enough to accurately predict her response to the situation. And, honestly, it's not that hard to figure that the first thing someone you've betrayed will ask you, if they have a chance, is "why?". She also knows Kovacs well, and knows that Quell and Kovacs were similar and had a bond and were likely to think similarly. In short, the answer may be Jack's comment: she was lying/improvising to suit her own ends.
Second possibility: in a subsequent episode she claims that just before the explosion she backed up Quell's stack, and only she knows where to find it. If we assume this is true, then Rei could have easily found out what happened via that stack. By interrogating her in a sleeve, say. Or we might believe she really did back up herself, too, between answering Quell and the explosion if we take her "just before the explosion" line quite seriously. Kovacs still entertains the possibility Rei was telling the truth at the end of the story, or at least that Quell's stack really might have survived somehow.