Does the "bi" in bilinear and biorthogonal mean different things? Does the "bi" in bilinear and biorthogonal mean different things? Bilinear seems to be linear from both left and right sides but biorthogonal means the product is zero sometimes instead of always?

Bi essentially just means two.
In bilinear it means linear in two arguments. In biorthogonal it means two families of vectors are orthogonal in respect to one another (but neither must be orthogonal in respect to itself). In binomial it means that there are two summands within the power, i.e. $(a+b)^n$. In binary it means that there are two possible digits, $0$ and $1$. 
