Can “You shall go to piano class” have 3 different meanings depending on the speaker or the context?
The definition of “shall” in the dictionary is “used to say that (1)something certainly will or (2)must happen, or that (3)you are determined that something will happen.
1) If the speaker has the power to influence or control events and circumstances: (something certainly will happen)In this case, the kid cannot afford the piano lesson and the speaker is the owner of the private institution of piano lesson.
"Your attendance at the piano class is fixed because nothing prevents you." (like “You shall go to the ball, Cinderella” by fairy godmother)
2) If the speaker is a very old-fashioned parent or very formal: (something must happen = command)“Go to piano lesson” or “You will go to piano lesson” (contemporary usage)
3) If the speaker does not have the power but is determined that something will happen: In this case, he wants his kid to be able to go to piano lesson even though he does not have enough money right now: (He is determined that his kid will have a piano lesson)“I am determined (firmly decide / promise /will exert my ability) to let you have a piano lesson”
Let me have another example of “You shall die.”It goes to (1) but I think it can also go to (3) where it means “I am determined to do everything not to let you live even though I do not have power to kill or control your life.”
Am I right with 3 typical usage?
Your explanations are very clear and show an accurate understanding of the various meanings and usage of the word shall. When thinking about these types of sentences though, I would caution you about "making up a story" about the sentence -- it could mislead your understanding if you are trying to learn how to accurately translate. In general, in English "shall" can be used where "will" could be used. As you noted in #2 above, there is a connotation of a more emphatic intent.
The other way shall is used in formal English is with 1st person questions. (3rd person questions use "will.") Here are a couple of examples:
1. Shall I make dinner for the family?
2. Shall we go out to eat at the local diner?
3. Will he make dinner for the family?
4. Will they go out to eat at the local diner?