Is the husband's saying to his wife in a moment of anger, "By God, I do not love you, and by God, I do not want you," considered an oath of divorce or an oath that necessitates expiation for breaking an oath, especially since they have agreed to start a new chapter?

This statement is not considered a divorce, nor does it require an expiation for an oath, because he swore to confirm an existing matter. Both spouses must fear Allah and be good to each other. The husband is commanded to be gentle, patient, and to uphold the Prophet's ﷺ injunction: "Fear Allah concerning women," and "Treat women kindly." The wife is commanded to obey her husband in matters that do not involve disobedience to Allah. The husband's right is great, and the Prophet ﷺ said: "If I were to command anyone to prostrate to anyone, I would have commanded the woman to prostrate to her husband," and "If a woman prays her five [daily prayers], fasts her month [of Ramadan], guards her chastity, and obeys her husband, she will be told, 'Enter Paradise from whichever of its gates you wish,'" and "There is no man who calls his wife to his bed, and she refuses him, but that the One in heaven is displeased with her until he [the husband] is pleased with her." Let each of them fear Allah concerning the other so that they may live a happy life.