Chapter 14 of Prenowitz and Jordan, Basic Concepts of Geometry, begins, "In this chapter a tentative treatment of congruence is given based on a proposal of G. D. Birkhoff (1884-1944) that the real number system should be assumed in the treatment of Euclidean geometry at an elementary level. Birkhoff's development was modified and simplified by the School Mathematics Study Group. Our treatment is an adaptation of theirs and assumes a modification of their Ruler Postulate employed by MacLane."
References are given to Birkhoff and Beatley, to the School Mathematics Study Group textbook Geometry (Yale U. Press, 1961), and to S. MacLane, Metric postulates for plane geometry, American Mathematical Monthly 66 (1959) 543-555.
There is no discussion of projective or hyperbolic geometry in Chapter 14 (but there is an extensive discussion of hyperbolic geometry in earlier chapters).
You might also be interested in Moise, Elementary Geometry From An Advanced Standpoint. In Chapter 8 you find out that the way he has been presenting plane geometry "is not the classical one. It was proposed in the early 1930's by G. D. Birkhoff, and has only recently become popular." In later chapters he does hyperbolic geometry, but I don't know whether he follows the Birkhoff path. I see no mention of projective geometry in this book.