Is it true that the height of a regular pyramid is the height of a lateral face?

No, it is not true that the height of a regular pyramid is the height of a lateral face. The height of a regular pyramid is defined as the perpendicular distance from the apex (top vertex) to the base. In contrast, the height of a lateral face (triangular face) is the perpendicular distance from the apex to the base of that triangular face, which is not necessarily the same as the pyramid's overall height.