Talaq is divorce. The man can divorce his wife at any moment, without any witnesses and without even her knowledge of it. Mahr is always the woman's right, and the custom that you
only receive your mahr in case of divorce was not the intention. I'm not even sure it is completely permissible, but it was a long time ago I sat with my nose deep down in my fiqh texts...
As for khula', it is not a divorce on the initiative of the woman. It is a request for separation. The woman can ask the man to divorce her, but in case he is unwilling she can take up the issue with a judge or whoever was in charge of legal issues back then (Muhammad was...). It's not like she automatically gets it, the husband can ask for the mahr (he doesn't have to), but he can also ask for more than just the mahr. It is basically the woman buying herself free if the judge allows it. He can always deny her request, even though there are no legal restrictions to the reason why the woman wants to separate. But as we all now, reality isn't always like it is portrayed in idealized texts and books. The legal text they use for khula' is the hadith where a woman came to the Prophet and asked for separation from her husband, because "she didn't like him" , and the Prophet said she was free to go if she returned the mahr. Which means, she doesn't have to have a "legitimate reason" as she obliged to have when she asks for divorce where the judge decides the matter. For example, if the husband fails in some of his obligations towards his wife.
So you have talaq, you have khula', but you also have separation between the spouses where the judge grants the woman the right despite the husband's protests (and she still keeps her mahr)...
But in all honesty, who cares about the mahr? The mahr is only of substantial worth if the woman and/and in some cases or the man comes from a wealthy family. In most cases the woman doesn't even receive a mahr despite them agreeing on one. Because the husbands simply doesn't give her any, and the woman silently agrees to forfeit it. May Allah reward her
This is the reality of women, of all races and creed, that economic independence is the key for any kind of female emancipation. That is also why all patriarchal structures and systems object to women's economic independence on the grounds of "moral issues".
Topic: Divorce in Islam