I think there is no way to distinguish between пиво and пива here in the audio since in Russian they sound exactly the same. The meaning "Do you want beer" is also legitimate I think, for example, if there is a choice of different drinks.
I second that (native speaker).
There is no way to tell them apart when pronounced properly in Russian, full stop - i.e., not just in this audio. With the exception of few very local dialects, an unstressed "o" is basically a schwa sound (https://en.wikipedia.org/wiki/Schwa), same as an unstressed "a".
And your interpretation of the difference between "Ты хочешь пива?" (genitive) and "Ты хочешь пиво?" (accusative) is absolutely correct as well. Btw, the latter could also mean "Would you like the beer" - I could be handing you a bottle while asking this question.
Because this gives the meaning of "some beer" (a quantity, but not strictly defined) as opposed to "the beer". It's common with liquids, also: "ты хочешь воды?"
So does this mean that my translation "Do you want a beer" is incorrect here? Because it was rejected now.