The audio really isn't clear on this one. I listened to the slower one a few times and the v sounded like an m, like "un main" or "un min"
Un pain / un vin ... sounds the same to the untrained ear on regular laptop speakers. We lose hearts for not having Hi Def sound systems :)
Plus "un vin" sounds stupid to me. "Un verre du vin" makes much more sense.
Keep up the great work guys ;)
On second part of your question: "un vin" means an indefinite wine, which can be red of white, Bordeaux or Bourgogne, etc.
"un verre du vin" would be correct in a sentence like "donne-moi un verre DU vin qui est sur la table" (give me a glass OF THE wine that is on the table).
"a glass of wine" is "un verre de vin" because it is a portion of an uncountable thing.
Please to Google/Translation and type "un vin" in the French-English section, then click on the loud speaker, she says it perfectly. Then practice! You may even type "vin, vent, vont" and practice the differences between nasal sounds.