A beverage is something that you drink. A "drink" often refers to a liquid with alcohol in it depending on the context. People will request "a drink of water" but more often they will ask for "a bottle of water" to avoid drinking "tap water". A "soft drink" is one without alcohol.

I want to drink usually means an alcoholic drink and beverages are any type of drink ,

Beverage is often used in the industry, rather than in every day speech, although people might say, "care for a beverage?" as an alternative way to say "would you like a drink?"