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Do we have extra-biblical records of more than one census under Quirinius?
No. We don't have any independent records of ANY Roman census of Judaea prior to 6 AD, or any record of Quirinius holding the governorship of Syria (or any other position that Luke might have meant) prior to 6 AD.
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Luke pointedly refers to the "first" census under Quirinius, implying that Quirinius held at least one later census.
I think that you are misreading it. I believe that Luke meant that it was the first Roman census of Judaea - which was held under Quirinius. Because Judaea was a nominally independent client state up until 6 AD when it was absorbed into the Roman Empire there would have been no earlier census. (And if there had been, Jospehus - who was very interested in the relationship between the Jews and the Romans - should have recorded it)
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Why do you assume that the 6 AD census was the first one under Quirinius, and not the second or a later one?
I don't simply assume it. I conclude it, given that there are good reasons to reject the idea of any earlier Roman census (under any governor) - and no reason to place Quirinius as governor of Syria prior to 6 AD.