I like wine, but I am drinking milk. — J'aime le vin, mais je bois du lait.
Is it correct to say "J'aime du vin, mais je bois du lait", because "le vin" sounds more specific to "this wine" rather than wine in general?
If you say "je bois du vin", that means I'm drinking wine. Or, I'm drinking some wine, but you don't need to say "some" in English. If you say "je bois le vin", it means you're drinking THE wine, or all of the wine. In this case we know what wine we're talking about, as opposed to all the wine in the world.
So you say "j'aime le vin", which means you like wine in general, but you are drinking some milk, "du lait", not "le lait" which would mean you're drinking all the milk. If you have a roommate the opens the fridge and there is no more milk, it's all gone, he/she would say "est-ce que tu a bu le lait"? We know what milk we're talking about. It's finite. But if you say "j'ai bu du lait", it means I drank some milk.
Check the "Tips & Notes" for the Food lesson. This rule is explained very well there. Basically, one of the times that you MUST use the definite article in French (le/la/les) is before the direct object (in this sentence, "vin") of a verb of appreciation (in this sentence, "aimer"). Hope that helps.