Is blood that is discharged after seeing the white discharge and before thirteen days have passed considered menstrual blood, which necessitates abstaining from prayer and fasting, or is it irregular bleeding that does not necessitate that, and is it obligatory to make up for the fasting days during which it occurred?

The last blood the questioner saw at the beginning of Ramadan, which continued, is istihada (vaginal bleeding outside of menstruation), not hayd (menstruation). This is because it cannot be combined with what preceded it to form a single menstrual period, nor can it be considered a separate menstrual period. What the questioner did by considering it dam fasad (irregular bleeding) is correct, and she is not required to make up the prayers for the days it occurred.

If the bleeding continues, the questioner should observe her customary period if she has one, and the remainder is istihada. If she does not have a customary period, she should rely on differentiation (identifying menstrual blood by its characteristics). If differentiation is not possible, she should estimate six or seven days, considering them as hayd, and anything beyond that is istihada.