There is no consensus among scholars about the 'Idah of a woman who asked for Khul'a. Note that count for this waiting period only starts once the verdict is final and can't be made void!
This difference comes from two different views: Is khul'a (as a separation) equal to talaq (divorce) or is it equal to faskh (declaring a marriage as invalid).
Imam Ahmad and others have adapted the later view. Among those who hold this (2nd) view you may find Imam a-Shafi'i in his ancient madhab, ibn 'Abbas, 'Othman, 'Ali, 'Ikrimah, Tawoos, ibn Taymiyyah and ibn al-Qayyim.
So on one side we have the main View of Imam Ahmad and the view ibn al-Qayyim defended based on his qualification of the hadith which has been narrated in sunan abi Dawod, Jami' at-Tirmdihi and sunan an-Nasai.
While the majority of scholars consider that the waiting period of a woman who has asked for separation (by khul'a) is the one of a woman who has got divorce based on the Quran Verses:
Divorced women remain in waiting for three periods ...2:228
And those who no longer expect menstruation among your women - if you doubt, then their period is three months, and those who have not menstruated. And for those who are pregnant, their term is until they give birth ... 65:4
And Imam Malik also added some statements of sahaba and tabi'yn like here, here and here.
I found a fatwa supporting the first view and an article supporting the later.
Note that the Article has explained why this is a stronger view: The given hadith has several issues in the content and sanad (according to al-Qurtobi) and the sahih hadith about the khul'a of the wife Thabet ibn Qays doesn't mention a 'idah at all, but supports the view of talaq. As it clearly says that our Prophet asked Thabet to divorce his wife! You can read the proof of the other view in the fatwa above.