All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
If the thing upon which you rendered your vow conditional has actually occurred, then it is incumbent upon you to deliver on your vow in the way that you have mentioned in your question. We would like to draw your attention to some points: Firstly, you should fast the month of Ramadan with the intention of observing obligatory fasting. It is impermissible for you to fast in Ramadan with the intention of fulfilling your vow combined with that of the obligatory fast of that month. Secondly, you should not fast on the two 'Eed days, the days of Tashreeq, and days of menstruation and post-partum bleeding. Thirdly, if you have specified a particular time to start fasting these ten years, for example, you may have vowed to fast ten successive years starting from the following year, or starting from the month of Rabee‘, or starting tomorrow. In this case, it is not incumbent on you to make up for the days on which you are obliged not to observe fast, i.e. the 'Eed days, days of menstruation, or post-partum bleeding. You are not obliged either to make up for the fast-days of Ramadan that you fasted. However, if you did not specify a particular time to start the fast, then you are obliged to make up for the days on which you were obliged not to observe fast due to illness or post-partum bleeding, or the 'Eed days. You are also obliged to make up for the days in Ramadan that you fasted with the intention of observing obligatory fasting. This is the opinion of the majority of Muslim jurists. Fourthly, if you fail to fast on any day of those ten successive years without a Sharee'ah-approved excuse, then you should start fasting all over again (since you had vowed to fast the ten years successively). As for the Hadeeth that you mentioned in your question, it is irrelevant to your vow.
Allaah Knows best.