Did Quirinius have two terms?

Mainstream scholars claim that Quirinius only served one term as governor of Syria. This has caused difficulties for Biblical inerrantists, who must show that Jesus was born during the reign of Herod the Great (Matt 2:1) and the term of Quirinius (Luke 2:2). But historians have concluded that Herod died around 4 BCE and Quirinius's census is usually dated at around 6-7 CE. One of the most creatives responses to this difficulty is to claim that Quirinius had two terms.

Arguments made by apologists I've read only show that it's possible that Quirinius served twice, or that he must have served twice because Inerrancy requires it. Indeed, I've seen internet apologists assume this as fact when it's very controversial (perhaps even wrong). But I'm not a historian and I'm willing to be set straight. What good evidence are there for this argument?

There is a different argument used by inerrantists which claim that the word πρωτη (prōtey/prōtos) in Luke 2:2 should be translated as "before", not "first". My Greek is still at the novice level so I can't judge the accuracy of the translation, but I think this would be a better avenue to defend inerrancy than to make up history. Philologists might beg to differ.