His wife asked him for a divorce, so he gave her khula‘ even though she refused it. Is their separation regarded as khula‘ or talaaq?

With regard to your saying to your wife: “I agree to liberate you…”, if you intended thereby simply to agree to the idea of separation in principle, and that you would not prevent her leaving, then this does not count as anything. But if you intended thereby to initiate the separation, then it does count as separation and is a revocable talaaq, not khula‘. 

That is because khula‘ is a contract like any other, in which there must be a proposal and acceptance. It says in Durar al-Hukkaam Sharh Ghurar al-Ahkaam (1/389): It requires a proposal and acceptance like any other contract. End quote. 

You said that your wife did not agree to the khula‘. Based on that, no khula‘ has taken place, because the condition of acceptance on the part of the wife was not fulfilled. 

It says in al-Insaaf by al-Mirdaawi (22/44): 

Abu Bakr said: It was narrated from Abu ‘Abdullah that khula‘ is that which is initiated by women. If it is initiated by the men, then there is no difference of opinion concerning the fact that it is a revocable talaaq in which he has the right to take his wife back, and it is not an annulment of the marriage (khula‘). End quote. 

As no khula‘ has taken place, then what has taken place a revocable talaaq. You say that this happened after having intercourse with her during a period of purity. Therefore it is a talaaq bid‘i (innovated divorce), which is a kind of divorce that is prohibited, and whether it counts as such or not is a matter concerning which there is a difference of scholarly opinion. We have explained previously in a number of fatwas that the view that is more likely to be correct, and the view on which fatwas on our website are based, is that it does not count as such. Please see fatwa no. 178554

And Allah knows best.