Is it considered backbiting for a wife to tell the girl her husband is speaking with what he said about her, and to warn her about his lack of seriousness in marrying her?

What you did—conveying the news to that woman—is considered backbiting (namimah) and a disclosure of someone else's secret. Backbiting is the divulgence of a secret that one dislikes being revealed, and it is forbidden and blameworthy. The dispraise of backbiting is mentioned in the Holy Quran in His (Allah's) saying: {a slanderer, going about with calumnies} (Al-Qalam:11). And in the noble Hadith: "A backbiter will not enter Paradise," and that one of those tormented in the grave was a backbiter. Therefore, you must repent and seek forgiveness for this sin.