There is, of course, no distinction between the words "bisector of an angle" and "angle bisector". The only difference between the two phrases you give are the words "of a triangle". To have a "bisector of an angle" (or "angle bisector") you only have to have an angle- there may be no "triangle" involved. To have an "angle bisector of a triangle" (or "bisector of an angle of a triangle") you have to have a triangle and, since a triangle has three angles, either specify which angle or make it clear that it could be any one of the angles. At that point you are talking about the bisector of that specific angle.
Well there you have it. For that particular author the phrase angle bisector of a triangle mean a certain line segment which is a subset if the general angular bisector, the coplanar ray which makes angles of equal measure with the sides of an angle.
As a line segment the angle bisector of a triangle has a finite length unlike a ray. For some authors that is an important distinction.