I think it's that "I want to have a cup of coffee" includes all the process of "having a cup of coffee". (e.g. drink a bit --> rest a while --> drink again)
cause most of the time when people have a cup of coffee, they don't just drink the whole thing at once, but they drink a bit and enjoy time passing or have a little chat with someone else then drink again. so it takes time.
but "drink a cup of coffee" indicates only the action of drinking it. rather an instant thing.
It's just my opinion. but it makes sense.