As far as I can tell, both of these are perfectly sufficient definitions for a line segment $\overline{AB}$. The book I am using defines them the latter way, but is the first definition also equivalent?
If so how would I prove that these sets are in fact the same?
In other words how would I prove $$\overline{AB}:=\overrightarrow{AB}\cap \overrightarrow{BA} = \{A,B\}\cup \{X: A-X-B\}$$
EDIT: $A-X-B$ reads $X$ is between $A$ & $B$.
In other words, $A$, $B$, & $C$ are collinear and $\overrightarrow{XA}\cap \overrightarrow{XB} = \{X\}$
According to my definitions
$\overrightarrow{AB}=\{A,B\}\cup\{X\colon A-X-B \vee A-B-X\}$ and $\overline{AB}=\{A,B\}\cup\{X\colon A-X-B\}$.
Hence the inclusion $\overline{AB}\subset\overrightarrow{AB}\cap\overrightarrow{BA}$ is obvious.
For the opposite inclusion you only need to know that
$P-Q-R \implies R-Q-P$ and $P-Q-R \implies \neg Q-P-R$,
in other words betweenness is symmetric and at most one of the three points is between the other two. Both these propertries are usually taken as axioms in axiomatic systems.