How is it that both Jerusalem (II Sam. 5:6-7) and Bethlehem (Lk. 2:4) are called "the city of David"?

In the Old Testament, the term always refers to Jerusalem. The city was viewed as belonging to David by right of conquest. Since he conquered the city, it became "David's city" or the "the city of David."

In Luke 2:4, being written in a different language (Greek, not Hebrew), the term "the city of David" is not emphasizing ownership, but origin. David was born in Bethlehem, therefore, Bethlehem was known as "the city of David," and that is the emphasis in this passage.