In (1) you are giving more of a direct command. You are saying that you want someone to play the piano, insinuating that there is a piano present in the setting, and that you would like that someone to play it in that moment.
In (2), it is similar to (1). You can take the statement the same way, where you presume there is a piano in the setting, and you'd like someone to play it for you in that moment. You can also take this situation as a more general statement, like when a parent tells you that they'd like you to start taking piano lessons, because they want you to learn to play the piano. In that case, (2) makes sense too. Because you say "the piano" instead of "piano" you can direct it to a specific piano in the setting.
In (3), I don't really hear it that often. The way I picture it in my mind is that you are in an instruments shop, and you are choosing a piano to buy, and therefore you need to play one to see the difference. It is a more general statement, and although it makes sense, (1) and (2) are more common.
Hope this helps. Let me know if there's more clarification needed!