"Instead of ^ (which is commonly know as "to the power of") use this ↑.
They are not the same thing."
Incorrect. When it has been specified that "^" is being used to stand for Knuth's up-arrow, then yes, it becomes the same thing.
Unless a computer keyboard has the up-arrow, it's necessary to substitute something else for it.
A good reason why "^" is a good substitute is the fact that one up-arrow means the same as "^": Exponentiation.
Therefore, "^" is the natural and obvious substitute for the up-arrow, if the computer keyboard doesn't have the up-arrow.