All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
Divorcing the wife after her request for divorce is not Khul’, but it is a revocable divorce.
Fataawa Daar Al-Iftaa’ Al-Misriyyah reads:
"Every divorce takes place as a revocable divorce except for the one that completes three divorces, the divorce before the consummation of the marriage, and the divorce in return for money." [End of quote]
Accordingly, what happened between you and your wife is a divorce. Nonetheless, since it is not the third divorce, then your taking back of your wife is valid, and your marriage is still ongoing and is not invalidated.
For more benefit on an irrevocable divorce, please refer to Fataawa 384739, 381389, 128967, and 116098.
For more benefit on divorce given in anger, please refer to Fataawa 220696, 243088, 322960, 388225, 395773, 121374, 126164, and 245707.
Allah Knows best.