Must a woman re-perform the prayers for the day on which her menstrual blood was delayed in exiting and remained inside the vagina, and is a woman considered to be menstruating merely by seeing blood upon inserting a cotton swab, or must the blood flow and exit?

The correct view is that the ruling of menses is not established until the blood exits the vulva. So, if a woman feels the menses moving, or if the blood is inside her vulva but has not emerged outside, it is not considered menstruation and does not affect her fast, based on the hadith of Umm Sulaym: "Yes, if she sees the water." As long as the sight of blood inside the vulva has not emerged outside, the ruling of menses is not established, and she must make up the prayers she missed.