Is what the questioner did correct by considering the blood discharged after the first purification as istihada (non-menstrual vaginal bleeding) and her subsequent actions based on that? And is the brownish discharge she saw after purification considered a sign that menstruation has not ended, and what should she do regarding prayer and fasting in this situation?

The returning blood from the previous menses is considered menstruation, unless it exceeds 15 days when combined with the previous blood and the period of purity in between. If it exceeds that, it is considered istihadah (non-menstrual vaginal bleeding). Any blood seen after its cessation necessitates re-performing ghusl (ritual bath). The fast of the day on which blood was seen after Maghrib (sunset) is valid. Brownish discharge after Fajr (dawn) is not considered menstruation. One must continue fasting and praying, as the woman in this situation is ritually pure.