Her husband divorced her (talaaq) then took her back, then she asked him for a second and third talaaq, and is asking whether this is khul‘ or talaaq
It is not permissible for the wife to ask for a divorce (talaaq) from her husband without a valid shar‘i reason, because the Prophet (blessings and peace of Allah be upon him) said: “Any woman who asks her husband for a divorce when it is not absolutely necessary, the fragrance of Paradise will be forbidden to her.”
Narrated by Imam Ahmad, 21874; and Ibn Maajah, 2055. What is meant by the phrase “when it is not absolutely necessary” is without there being any hardship which makes it necessary to resort to talaaq (as-Sindi’s commentary on Ibn Maajah).
What has happened is that your husband divorced you with the first talaaq, then took you back; then he divorced you once again, based on your request. Then you asked him for another talaaq after that, so he gave you another talaaq. In this case only two talaaqs have definitely taken place, namely the first and second talaaq. The second talaaq cannot be ruled out because of what your husband said about not wanting it to count as such, and that he only spoke the words to go along with your wishes, because if the husband utters the word of talaaq, knowing what it means and not being forced to say it, then the talaaq counts as such whether the husband really intended it or not. This has been discussed previously in fatwa no. 171398.
With regard to the third talaaq, if your husband had not taken you back after the second talaaq, then there is a difference of scholarly opinion as to whether the third divorce counts as such. Some of the scholars said that it does count as such and others said that it does not. The more correct view is that it does not count as such, as we have previously explained in fatwa no. 126549.
But if your husband had taken your back following the second talaaq, then he issued a third talaaq to you, then it does count as such according to scholarly consensus.
With regard to your question about what happened and whether it is a talaaq or khul‘, the answer is that it is a talaaq because it took place with the word of talaaq and without any compensation (from the wife).
In fatwa no. 126444 we have previously explained that in the case of khul‘ there must be compensation. Any separation between husband and wife that happens in return for compensation is khul‘, even if it is done by uttering the word of talaaq, according to the correct scholarly opinion.
And Allah knows best.