I translated this incorrectly as you did.
I think the problem is that would like is a conditional tense of to want in English. In Italian the conjugation for the verb volere in that case would be vorresti instead of vuoi.
Side note: Since it is so commonly interchangeable in English, I never really considered that it was a different tense of the same verb.
While I agree that in English it would be more proper to translate this as "what would you like," I think in this case for the sake of learning it is important to differentiate "cosa vuoi" from something like "cosa ti piacerebbe" which is what "what would you like" would directly translate to.
True, but in English if we're talking to family or friends to ask "do you want...?" is perfectly normal. It would seem rude if you were talking to a stranger, but since the familiar form "tu" is used in this sentence, don't see a problem with it. I may be wrong, but I think just to use "do you want" in the polite "Lei" form, wouldn't be particularly rude either.