Is the brown discharge that appears after menstruation stops, and after performing ghusl with the intention of fasting in Ramadan, considered a continuation of menstruation, or does it not invalidate the fast and require ghusl?

The color you are seeing is yellowish and brownish discharge, which is considered menstruation if it occurs during the menstrual period and is connected to blood. What you are doing, by waiting until the yellowish and brownish discharge stops and you see the clearness, is correct; this is based on the saying of Aisha, may Allah be pleased with her: "Do not rush until you see the white discharge." If these discharges continue after dawn on that day, your fast is not valid, and you must make it up. You must also perform ghusl again after the discharge stops and you see purity.