They are referring to what is called a conical pendulum in the first part. Such a pendulum does not swing back and forth. It moves in a circle at constant speed with the motion forming a cone in which the angle theta subtended by the string to the vertical is a constant.

It is difficult to explain without figures. Do look up the derivation of formula for the period and compare it with that of a simple pendulum.

For a conical pendulum, the period is given by

T = 2 pi SQRT (L cos theta / g).

For small theta cos theta ~ 1 so the period is the same.

As theta increases however, the preiod is smaller than that for a simple pendulum.