Is line segment intersection is different from intersection of lines ?
in one way they are the same, in another way they are different.
1- They are the same that an intersection is a point, and if that intersection point exist then the "things" intersect, if that point doesn't exist then they don't intersect. (that should be clear isn't it)
2- They differ that segments don't have to intersect if the extensions do intersect (because the intersection point is outside the two segments, for lines not to intersect they have to be parallel (or asymtopic) , segment can be non intersecting just because they don't .
Maybe best is only to remember 1,
2 is just a bit of background theory.
Hopes this helps