If the woman who has been divorced by khula’ is pregnant then her ‘iddah lasts until she gives birth, according to scholarly consensus. But if she is not pregnant, the scholars differed concerning her ‘iddah. Some scholars are of the opinion that it is sufficient for a woman divorced by khula’ to wait for one menstrual cycle, because of the hadith you referred to in your question.
However, other scholars said that she should wait for three menstrual cycles, because of the general meaning of the ayah:
“And divorced women shall wait (as regards their marriage) for three menstrual periods” (Al-Baqarah 2:228)
Those scholars who said that it is sufficient for a woman divorced by khula’ to wait for one menstrual cycle consider khul` as different from normal divorce and thus she can only wait for one menstrual cycle.
The other opinion considers khul` as a kind of divorce and thus the woman divorced by khul` should wait for three menstrual cycles based on the general meaning of the verse quoted above.
So, the difference here among Muslim jurists is regarding whether khul` is considered as the normal divorce or not.
Having clarified the above, we’d like to state that in case of a woman divorced by khu;` if she observes an ‘iddah of three menstrual cycles that will be more complete and will be on the safe side, and will avoid an area of scholarly dispute.