Well, I suppose that's one of the definitions; Hillel opted for the other one. :shrug:
I think that he wanted to emphasize the distinction between operators and functions; in your definition, there is no difference between "binary operator" and "function with two arguments". Of course, some languages limit what characters can be used in function names, but that is certainly not universally true.
A binary operator is simply one that takes two operands, their relative to positions do not matter one bit