Hilbert style axioms for Euclidean and/or hyperbolic geometry without reference to congruence? Hilbert's axioms from Grundlagen der Geometrie involve notions of incidence, between-ness, segment congruence and angle congruence.  
Consider the sub-theories of either Euclidean or hyperbolic geometry involving only the notions of incidence and between-ness.  Of course, a priori, notions of congruence may occur in proofs  (though not statements) of theorems solely about incidence and between-ness.  
As a matter of fact, consider any 2-dimensional compact convex body $B$, with every boundary point extreme, say, but $B$ not affinely equivalent to a round disk.  One can associate to $B$ an incidence geometry after the manner of Beltrami-Klein (with "lines" equal to interiors of chords).  Such a geometry will not have an isomorphism to hyperbolic or Euclidean incidence geometries even though Hilbert's axioms about incidence and between-ness will hold!  Indeed the incidence theory of such a quasi-Beltrami-Klein model actually determines $B$ up to an affine transformation.  (Has this fact been recorded in the literature?)
Main Question:  Can one formulate a finite, or at least an elegant set of incidence and between-ness axioms, extending Hilbert's, so as to capture the theory of Euclidean and/or hyperbolic incidence?
Added later: In light of Will Jagy's comment I'll sketch very briefly why I think the incidence geometry determines $B$ as much as it does, in case I'm obviously wrong.  Given $B$, one can decorate it in many way with three families of evenly spaced parallel chords with triple intersections wherever intersections occur.  The infinite of maximal configurations of this sort constitute approximations sufficient to recover $B$ up to affine. 
 A: The answer given to this question in the 20th century  is quite complex, and a summary of what was done can be found in  Victor Pambuccian, The axiomatics of ordered geometry
I. Ordered incidence spaces, Expositiones Mathematicae 29 (2011) 24–66.
A: Would you consider a positive answer for the Euclidean case being given by adding to the incidence and betweenness axioms the axioms for an affine plane with Desargues' configuration plus a completeness axiom along the lines of Dedekind's Axiom?  These can be formulated purely in terms of incidence and betweenness and will characterize the affine plane based on the reals, I believe.
I don't know about the hyperbolic plane, but I suspect that you could probably define ideal and ultra-ideal points (adding incidence and betweenness axioms as necessary to make sure that these points are well-defined), add the axioms that would make the result be the Euclidean affine plane and then use some kind of polarity axiom to characterize the hyperbolic plane.
Have you consulted Greenberg's book?  I don't have a copy handy, but I suspect that his references would lead you to a place where these questions are addressed.
