Postulate 1 doesn't define a straight line segment. It tells you that it's possible to connect two points with a straight line segment. A definition would tell you whether or not something qualified as a straight line segment.
The definition appears to be given, appropriately enough, in the "definitions" section that precedes the postulates, which says "a straight line is one which lies evenly with the points on itself". (As translated in https://farside.ph.utexas.edu/books/Euclid/Elements.pdf )
But although that is indeed how people recognize lines, it's not so easy to define what it means.
