Byzantine Mosaic of Mary and Joseph at the Census

I have read on several occasions that the author of the Third Gospel was mistaken when he wrote that there was a census taken by Caesar Augustus saying, “This was the first census taken while Quirinius was governor of Syria.” If this is what he said he was mistaken as follows:

(1) Quirinius was appointed governor of Syria in 6 CE.

(2) Herod the Great, who is mentioned in 1.5, died about a decade before Quirinius was governor of Syria.

We must ask if this is what the author was saying. The Greek text reads like this:

αὕτη ἀπογραφὴ πρώτη ἐγένετο ἡγεμονεύοντος τῆς Συρίας Κυρηνίου.

Most have translated ἀπογραφὴ πρώτη as the “first census”. This would mean the author thought Quirinius conducted two or more censuses and Jesus was born around the time of the first one. This would put Jesus’ birth at contradictory times.

Others have suggested that ἀπογραφὴ πρώτη should be translated “the census before…” which makes πρώτη function as a preposition rather than as an adjective. If this is so than the author was not mistaken since he is dating the census as the one (seemingly knowable one) that was prior to Quirinius’ governing of Syria (also possibly suggesting that Quirinius must have been connected with an additional, later census).

What do you say? Do you think this is a legitimate grammatical possibility that clears the author of a historical error? Do you have any problems with this explanation?