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?