Is it valid to count the nine days that the questioner fasted, in which she intended to fast without specifying whether it was to make up for Ramadan or for a vow, as making up for Ramadan with the intention of completing the remaining days for the vow?

Fasting to make up for missed days (qada') and for a vow (nadhr (definition)) requires specifying the intention. A general intention to fast is not sufficient. The nine days fasted by the questioner are not valid because she did not distinguish her intention for qada' from that for nadhr. Her obligation will not be fulfilled unless she intends, each day before dawn, to fast either for making up Ramadan (definition) or for the vow, until all the days are completed.