How can one differentiate between ovulation bleeding and menstrual bleeding, and what is the ruling on the different discharges (red, brown, white, transparent, yellow) that occur during the ovulation period? Is this considered a second menstrual cycle? What is the ruling on prayer and fasting in this case?

Any blood a woman sees at a time when it can be menstruation is considered menstruation. What the questioner saw on the thirteenth and fifteenth days is considered menstruation, and she must perform ghusl after it ceases. As for the brownish discharge after that, it is not considered menstruation, and ghusl is not required for it, and her fasting and prayers are valid.