I'm reminded of W.A. Coppel's book which looks at these kinds of structures from a slightly different vantage point, namely closure systems. I can't actually find the book right now, but here's a quick synopsis of what's going on.

Given a set $X$ together with an arbitrary function $[\bullet,\bullet] : X^2 \rightarrow \mathcal{P}(X),$ let's call $A \subseteq X$ *closed* if and only if for all $a,b \in A$, we have that $[a,b] \subseteq A$. It's easy to show that an arbitrary intersection of closed sets is closed, or in other words the collection of closed sets forms a Moore family. This gives us a closure operator on $\mathcal{P}(X)$, which makes $(X,\mathcal{P}(X))$ into a closure system. This means that the collection of closed sets forms a complete lattice given by $$\bigwedge_{i \in I} C_i := \bigcap_{i \in I} C_i, \qquad \bigvee_{i \in I} C_i := \mathrm{cl}\left(\bigcup_{i \in I} C_i\right).$$

If we furthermore assume that $[a,a] = \{a\},$ which is true in an arbitrary metric space (though I'm not sure whether this follows from your axioms), then we can prove that singletons are closed sets. Thus we're allowed to consider the expression $\{a\} \vee \{b\}$ in the aforementioned lattice. This set clearly includes $[a,b]$, and I think your axioms probably imply that these sets are equal. Ergo the Moore family alone allows us to recover the original between-relation.

I think that if you take the first three axioms you've written down, and add in the axiom $[a,a] = \{a\},$ the resulting axiom list should provide a complete characterization of closure systems $X$ in which firstly, every singleton is closed, and secondly, for all $A \subseteq X$, we have that $A$ is closed if and only if for all $a,b \in A$, we have $\{a\} \vee \{b\} \subseteq A$. But you should check this explicitly rather than taking my word for it, because right now this is just a hunch.

I'll also remark that you might learn interesting things by looking at partially ordered sets and defining $[a,b] := \{x \in X : a \leq x \leq b\} \cup \{x \in X : b \leq x \leq a\}.$

Notes on coarse median spacesby Bowditch $\endgroup$Treelike structures arising from continua and convergence groupsand Adeleke and Neumann's MemoirRelations related to Betweenness. $\endgroup$4more comments