Short answer
– The most likely author of Luke 2:1, 4–5 is the same person traditionally credited with the Gospel of Luke — Luke, the physician, a Gentile and companion of the Apostle Paul. Evangelical Christians commonly accept Luke as the human author of both Luke and Acts and regard him as a careful, historically minded writer.

Why evangelicals identify Luke as the author
– Early church testimony: church fathers (e.g., Irenaeus, Clement of Alexandria, Tertullian, and later Eusebius) uniformly attribute the Gospel and Acts to Luke.
– New Testament corroboration: Paul names “Luke” in the epistles (Colossians 4:14 — “Luke, the beloved physician”; Philemon 1:24; 2 Timothy 4:11), and the “we” sections in Acts strongly suggest a companion who wrote from first‑hand knowledge of parts of Paul’s journeys.
– Internal unity: Luke and Acts form a coherent two‑volume work (both addressed to Theophilus and linked by vocabulary, style, theology, and arrangement), and readers from earliest times treated them as one authorial enterprise.

What evangelicals say about Luke as a writer and historian
– A careful historian: Evangelicals often emphasize Luke 1:1–4 where the author states his aim to give an “orderly account” based on eyewitness testimony and careful investigation. This supports confidence in Luke’s reliability for historical detail (e.g., naming political figures such as Caesar Augustus and Quirinius in Luke 2).
– Medical background and Gentile perspective: The descriptor “physician” and certain technical terms and medical-level detail in Luke’s Greek are cited to support the view that he was a well‑educated Gentile who wrote for both Jews and Gentiles.
– Pastoral and theological aims: Though historical, Luke’s narrative is also theological — highlighting fulfillment of prophecy (e.g., Jesus born in Bethlehem), God’s care for the lowly and outsiders, and the extension of salvation to the nations. Evangelicals therefore read Luke as both trustworthy history and faithful theological witness.

Date and setting (typical evangelical range)
– Many evangelical scholars date Luke‑Acts in the 50s–60s AD (some early; e.g., before Paul’s death), others later into the 60s–80s. A common conservative evangelical position places Luke’s writing in the 60s while he was still alive and active with Paul or shortly thereafter, but there is a range of views and modest uncertainty on precise dating.

Relevance for Luke 2:1, 4–5
– Those verses place Jesus’ birth within concrete Roman political events (a decree from Caesar Augustus, a census associated with Quirinius) — an example evangelicals use to argue that Luke intends to anchor the nativity in real history and that the author was attentive to historical detail.
– Evangelicals typically defend Luke’s historical accuracy and rely on him for theology and chronology of the infancy narratives, even as they recognize and discuss difficult or debated details (for example, how Luke’s census relates to extra‑biblical records of Quirinius).

Further reading (evangelical-friendly scholars)
– I. Howard Marshall, The Gospel of Luke (IGNT)
– F. F. Bruce, The Acts of the Apostles (NICNT)
– D. A. Carson, The Gospel According to John (for methodological parallels) and various essays on Gospel reliability
– Leon Morris, Studies in the Gospel of Luke

If you’d like, I can:
– Summarize the internal and external evidence for Luke’s authorship more fully;
– Explain how evangelicals reconcile Luke’s census reference with historical records;
– Provide a short devotional reflection on Luke 2:1, 4–5 from an evangelical perspective.