Is what I did correct, by not performing ghusl until I was certain of purity, given the presence of a brownish discharge (kudra) and clear fluid, and my doubt about the examination spot? And can the "white discharge" (al-qassah al-bayda') appear with kudra? And do I have to make up the prayers from that day?

It is clear that the questioner suffers from waswasah (obsessive whispering) and experiences disturbances in discharge due to using menstrual cycle regulating pills, which might cause the yellowish discharge (ṣufrah) to persist throughout the month. Purity was attained upon seeing the white discharge (al-qaṣṣah al-bayḍāʾ), and the continued presence of brownish discharge (al-kudrah) after that is not to be considered. She should have promptly performed ghusl (definition) (ritual bath) and prayed after seeing the white discharge. She must make up the prayers she missed on the day purity was attained. Purity is ascertained by wiping the outer part of the vaginal opening with a tissue and observing what appears when sitting. She must disregard these obsessive whisperings (waswasah).