I am not sure how vinegar was produced during the blessed time of Rasulullah Sallallahu Alaihe Wa Sallam but it is crystal clear that it existed during that time and it was one of the most favorite dishes of the Holy Prophet Sallallah Alaihe Wa Sallam as he used to say "what an excellent condiment is vinegar".
The Hadith that you have mentioned in your question is from Sahih Muslim and all the Ahadith in Sahih Muslim are authentic and correct without any doubt. This leads to the conclusion that the Hadith quoted in the question is authentic.
Based on the Hadith above, the four famous Imams of Fiqh have differed in their opinions as to whether making vinegar from wine is allowed or not?
Before I mentioned the difference of opinion among the four Imams on this issue, it is important to understand that if the wine turns into vinegar by its self then it is permissible and Halal according to all the four Imams. However, if someone turns wine into vinegar with his effort and action then this is an issue wherein the Fuqaha have differed in their respected opinions.
Now as to whether this practice of turning wine into vinegar is allowed or not? There are two opinions among the Fuqaha:
1) It is allowed and the vinegar is pure and Halal. This is the opinion of Imam Abu Hanifah, Imam Awzai, Imam Laith and in one narration it is also the opinion of Imam Malik.
2) It is not allowed and the vinegar produced in this manner is not Halal and pure. This is the opinion of Imam Shafi, Imam Ahmad and Imam Malik in one narration. The well known opinion of Imam Malik is that the process of turning wine into vinegar its self is dislike but the product (vinegar) produced in this manner would be Halal and pure.
The obvious proof for those Fuqaha who believe producing vinegar from wine is not Halal is the Hadith mentioned in Sahih Muslim that is also quoted in the question.
The claim of those Fuqaha who believe that vinegar produced from wine is Halal and pure is also based on many Ahadith mentioned in various collections of Hadith.
According to those Fuqaha, the Hadith that prohibits turning wine into vinegar is about the early period of prohibition when wine was declared forbidden in Madinah. In order to emphasize upon the prohibition and to allow the Muslims to understand that wine is prohibited clearly forever, many preventive step were taken by the Prophet Sallallahu Alaihe Wa Sallam. Among the preventive step that were taken regarding the prohibition of wine was that even the pot in which wine was produced or kept were ordered to be broken and no one was allowed to even keep those pots and vessels. Later this was relaxed when satisfaction was received that people had fully recognized and understood the prohibition of wine.
Similarly, the issue of turning wine into vinegar was also prohibited because it would give people a reason to keep the wine which would not have helpful in allowing Muslims to fully understand and recognize that wine is prohibited in every way.
As mentioned above, those Fuqaha who believe that turning wine into vinegar is permissible present many authentic Ahadith to support their claim.
I will only mention one Hadith out of those Ahadith. Those who are interested to learn more about it should refer to the commentary of Sahih Muslim by Shaykh Mufti Muhammad Taqi Usmani (May Allah preserve him).
Imam Darul Qutni has mentioned the following Hadith in his famous collection of Ahadith called "Sunan Darel Qutni":
"Umm-e Salamah (R.A) said that we had a goat and the goat died. The Holy Prophet Sallallahu Alaihe Wa Sallam asked: what happened to the goat? we said: it has died. He said, why did you not benefit from its skin? we said, it is a carrion. He said, it (the skin) becomes pure by tanning just like the vinegar made from wine becomes halal"
(Takmilah Fathul Muslim vol 3 pg. 613)
Now the important question is that should a Muslim go and buy wine with the intention of turning it into vinegar? This question is a very difficult one because if turning vinegar from wine is allowed then why purchasing wine is not allowed with the intention of making vinegar from it.
The famous Hadith in which about 10 people are cursed with regard to the wine also includes buyer and seller of wine. Therefore, under no circumstance it would be permissible to purchase wine to make it vinegar.
Now one might think that what is the point of the Hadith mentioned above which says it is Halal to make vinegar from wine and how is it practical?
The answer to this is that this is applied in situations where one has received wine from someone as an inheritance or some other source. Now this wine that a Muslim has become owner of, can either be spilled down the drain or one could sell it. Spilling it is allowed and selling it is not allowed because of the Hadith. Another way to get rid of it could be destroying it by turning it into vinegar. This is called "transformation" (Tabdeel-e Mahiyat) so it was Haram when it was wine and it became Halal when it is turned into vinegar.
My respected teacher Shaykh Mufti Muammad Taqi Usmani writes in his famous book "Takmilah Fathul Mulhim":
"What appears to this humble servant is that " the Ulama have disliked the fact that a Muslim has wine in his possession which he needs to turn into vinegar...... What is prohibited is is that a Muslim should make an effort to get possession of wine to make it into vinegar but if someone received through inheritance or something similar then it is permissible to make it into vinegar. (Takmilah Fathul Muslim).
And Allah knows best.
Mufti Ikram ul Haq
Fatwa Center of America
17 Safar 1433/ January 12 2012