Does divorce take effect in the two aforementioned cases: either when the wife leaves the house after divorce was made contingent on her leaving, or when the husband says "You are divorced" in response to her request for divorce?

If a man tells his wife: "If you leave the house, you are divorced," with the intention of divorce and to affirm it, then upon her leaving, one divorce occurs. If he intended only to prevent her without intending divorce, then he must pay an expiation for breaking an oath (kaffarat yamin).

As for a man telling his wife: "You are divorced," in response to her saying, "If you are a man, divorce me," one divorce occurs whether he intended it or not, because explicit wording does not require intention. It is advisable to avoid using divorce pronouncements during arguments.