The fermentation process of wine was begun in the vats, but then it was taken from the vats and completed in jars or wineskins.
Actually, the fermentation process starts before the vats. Yeast is often on grapes in the field like white on rice, or literally, a white film of yeast on the grapes. Grapes damaged by picking are already food for yeast and various microbes. Fermentation and decay begins in earnest as soon as the grapes are pressed.
In just a couple of days, the grape juice will start becoming objectionably nasty, if alcohol fermentation hasn't been encouraged. (The Welch's juice you buy at the store will last longer, if left out, because it has been pasteurized and your home is relatively sterile.)
I appreciate Christians warning against alcohol abuse and against a lifestyle centered on drinking. But, it's shameless for Christians to misrepresent anything and demonstrate a total disregard for whether nor not what they're saying is true, and that includes wine in the Bible (verses in the Bible alleged to show non-alcoholic wine, claims wine in the Bible was mixed with water, claims ancient Romans used the word wine to mean just grape juice, etc.). It also doesn't reflect well on those Christians when the pretend they have no obligation to answer for their misrepresentations.
Hence, "new" wine was the wine already in its beginning stages of fermentation but was put in the wineskin to finish the process and age the wine. A.M.Wilson gives you the historical background in his book pages 251-270
In ancient times, fermentation would only come to a complete stop after a long while. They might even want to encourage people to drink it before the fermentation is completely done, because by that time, the wine might have deteriorated. Having some fermentation while while it's being bottled (put in wine skins) would also help extend shelf-life (the gas displacing air).
In all of its stages it is commonly called wine because that was the intent or ultimate design. Wilson provides numerous examples of this kind of language where there is no doubt wine not grape juice is the intent. Think about the terms "winepress" as surely what came out could only be grapejuice, but it is still called a "wine" press because the ultimate intent was to produce wine and therefore never called a "grapejuice press."
Jesus tells as that he drank alcohol, Luke 7:33–44. We know it was alcohol because Jesus also tells us that the Pharisees judged him for it (essentially calling him a wino). Even if it could be shown that "wine" is also used for mere grape juice, that wouldn't undo the numerous verses that speak positively about clearly alcoholic wine. But, there are no verses that apply the word wine to mere grape juice. The very few verses that might do that are, as you say, referring the the intended product, not the intermediate product. "Wine in the cluster" is a figure of speech, and it isn't even a reference to a literal cluster of grapes.
