Translation:A glass of white wine, please.
There is also a third form, "per cortesia"; they reflect three types of favors you can receive, to please you (fare un piacere), to help you (fare un favore) and to be kind to you (fare una cortesia). From this you might infer some sort of rule, but they're basically interchangeable according to personal style, although cortesia is generally felt as more polite and favore is probably the most common one.
"Cup" Guess you don't spend much time at the vineyards :) I strongly suggest you never use "cup." Unless! Your're stranded in a former Russian weapons factor, being chased by zombies and are about to die as they have you trapped and your mates find a bottle of vino blanco and all that's around are old plastic cups from a trade with China for faux Elvis pants, then it would be acceptable to ask for a "Cup of vino" :)
Uh, my parents were born in Italy and they use the Italian for "glass" for most drinking containers, and the word "cup" in English for most drinking containers. My whole family does this. A cup is a cup is a cup. Even if it isn't a cup. And a cup is a glass and a glass is a cup, regardless of whether or not it is a cup or made of glass. In fact, I've never known an Italian to actually make that distinction...
I notice that the pronunciation of "bicchiere" differs depending upon whether it comes from the single word hover or the whole sentence. In the entire sentence, the word has a definite "i" sound at the end, whereas in the single word it does not. Does this reflect real world usage? If so, is there a rule about when to pronounce the final vowel sound at the end of a word?
I see the writing exercise Italian listening to writing. Both fast and slow audio sound the same with "e" bicchiere.
I wouldn't recommend you to rely on the synthetic voice for inferring such pronunciation rules. Most likely the synthetic voice system is just wrong as it has been over and over again. But you are doing the right thing, asking the native Italians here :-)