Get has many different meanings. In the sentence, "get me a glass of water" it means "bring".
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Get me a glass of water? This is an imperative, ie an order. So replace the question mark with a period or an exclamation mark. But in most contexts, an order like this would sound rude. A more polite form is
eg Can you get me a glass of water, please?
"Get"means you can give them water any way possible even if you tell another person to give the water
Now " Give" means you specifically need to give them water , only you can give it to them , can't do it another way