Must the questioner make up for a day on which she intended to fast, thinking she was menstruating, but then it became clear that she was not, and on which she ingested something from the skin of her lips?

The questioner did not explain the reason for her assumption that the blood she saw was not menstrual blood. She must make up this day because she intentionally ate some skin from her lips, and because her intention was not firm for fasting; rather, she hesitated. One of the conditions for the validity of fasting is a firm intention. The default for blood seen by a woman is that it is menstrual blood, not istihaadah (non-menstrual bleeding). It is forbidden to eat anything from human skin.