Hilbert's Geometry Axioms

In Euclidean Geometry, Euclid's undefined objects were points and
lines, and there was a primitive relation between them. (i.e. A point
may be "on" a line.) In Hilbert's more rigorous system, he realized the
need to add the primitive notion of "between" (as well as "congruent").

Is it possible to define "line" using the notion of between-ness,
rather than leave "line" as undefined?

For example, suppose that the between-ness relation is denoted by A-B-C
(meaning that a point B is between points A and C). We accept
Hilbert's usual Axioms of Order regarding this relation.
Then we can define the term "collinear", by saying that a set S of
three or more points is collinear if, for any three points X, Y, Z in
S, we have X-Y-Z or Y-Z-X or Z-X-Y.

Then couldn't a "line" be defined as any maximal collinear set of
points?
What additional axiom(s) concerning the betweenness relation would be
needed, so that you could prove as a theorem that there is exactly one
line passing through two given distinct points? (Or maybe that would
be an axiom itself?)

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