Is a plane determined by two intersecting lines?

Yes.

To help visualize, consider this:

A 3-legged stool won't wobble on a flat surface (3 points of contact.)

The minimum number of points to define two intersecting lines is where the point where the lines intersect, and another unique point on each line. (3 points total) This is represented by where the 3-leg stool touches the flat surface. The flat surface represents any determinant plane.

If the lines did not intersect, you would require 4 separate points to define them. This is represented by 4-leg chairs, which by contrast often wobble (if only very slightly), since the extra point is not required to define the flat surface that it rests on (the determinant plane)

In essence, 'a plane is determined by two intersecting lines' is the same as saying 'a plane is determined by three uniquepoints', which are, in both cases true.