What is the difference, intuitively, between the directrix of a right circular cone and the directrix of a parabola?

Question

I recently watched a video that illustrated the conic sections. In the video, the narrator referred to the circumference of the circular base of the cone as the directrix. This is consistent with another description where the directrix of the cone was considered to be the plane that the generator rotating around the axis cuts. However, with respect to the conic section - parabola, the directrix is outside of the cone. What gives? These definitions of directrix don't seem to logically fit together. Is there a intuitive explanation that relates these?