Does divorce take place in the two aforementioned cases, given that the wife left the house despite the husband saying: "If you leave, you are divorced," and the second time the husband said to her: "If you leave, you are not under my responsibility"?

One divorce occurred, and a رجعة (reconciliation) took place. The second divorce did not occur, because you gave her permission to leave. Your wife must return to your home; neither she nor her family has the right to refuse, using the occurrence of divorce as a pretext. If a second divorce occurs, it is a revocable divorce (رجعي), and you have the right to take your wife back without her consent or the consent of her family. A woman who is revocably divorced must remain in her husband's home until her 'iddah (waiting period) expires.