The example that you gave, "I'm thirsty," means that the person wants to drink something that quenches (takes away or relieves) a feeling of thirst (needing a drink). It usually means a drink that helps relieve the feeling, such as water, juice, milk, and so forth. Normally, it doesn't refer to alcohol, although a cold beer is enjoyable sometimes. For your information, "thirsty" can be used to express a strong desire for something. For example, she was "thirsty" for success. But, your inquiry refers to a drink of some kind.

One can have thirst for anything really - alcohol, water, juice or satisfaction. A vampire can be thirsty for blood too.

It means he wants to drink. And "to drink" means to take a liquid into the mouth and swallow. Neither word is specific. To know what he wants to drink, you need to rely on context. Usually, it means any kind of drink--water, juice, a soft drink. If someone says "I'm thirsty," and you aren't outside a bar, they probably do not mean alcohol. However, the word "drink" itself can mean either "anything liquid you can put in your mouth and swallow," OR "an alcoholic beverage." For example, the related word "drunk" means "intoxicated with alcohol." "Don't drink and drive" means "don't drink alcohol and drive." So, in context, "thirsty" can mean "wanting alcohol." So you could say, for example, "during Prohibition, when alcohol was illegal in the United States, illegal 'speakeasies' sprung up to satisfy a thirsty public."