How can one differentiate between menstrual blood and istiḥāḍa (non-menstrual vaginal bleeding), and what are the signs of purity, in order to determine the times for prayer and fasting, given the prolonged bleeding of various colors?

What you see of red and black blood is menses. The yellow or brown discharge is menses if it is connected to the black blood or if it occurs during the habitual period; otherwise, it is not considered menses. If you see purity (tahara) by dryness after this discharge ceases or before it (if it was not menses), then you have become pure. The white discharge is not menses. All of this applies as long as the duration of the blood does not exceed fifteen days. If you see blood after thirteen days or more have passed since you saw purity, then you have become a menstruating woman again.