Does divorce take place if the wife says: "If we are going to continue like this, then it is better for each of us to stay at our parents' home," and then the husband says: "I spoke as you wished, and you bear the responsibility," and then the matter is settled in the same call?

If "كتب الكتاب" (documenting the marriage) refers only to the engagement, then nothing is binding upon you because there is no legally valid contract. However, if it refers to the legal marriage contract without consummation, then she is a wife whom divorce affects. Regarding the exchange of words, if she said: "If we are to remain in this situation, it is better that each of us stays in their parents' home," and you replied: "As you wish" or "As you please, with assuming responsibility," then if she did not intend divorce, nothing occurs. But if you intended divorce, then it is effective and irrevocable (ba'in) because it occurred before consummation, and she is not lawful for you except with a new contract. You must separate from her immediately if you do not renew the contract.