This is from a third grader. His example is the point where the hands on the clock meet. It's not pivot. Or "if you start with a dot and make two lines go out from it, on straight up and one to the left, what would the dot you started at be called?"
(There's no "angle" tag!)
On
What you are asking in the body of the question is different from what you are asking in the title. As answered by zz20s, the point where two segments (finite) or rays (infinite on one side) join to form an angle is called a vertex, but the point where two lines (infinite on both sides) meet is called the intersection of the two lines.
I believe it is called the vertex of an angle.