Must one make up for the day in which the questioner was unable to determine whether her menstruation began before or after sunset?

The basic principle is that the fast is valid. Doubt about the onset of menstruation before sunset does not invalidate the fast. Therefore, your fast is valid, and you are not required to make up this day, because certainty in the validity of the fast is only removed by an equally certain matter. Consequently, you are not liable for expiation for delaying the make-up. However, if you were a minor and had not reached puberty, neither fasting nor making it up would be obligatory for you.