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?**