Impure functions lack referential transparency, which is often sacrificed but kind of important to being a function - getting the same result every time you call it with the same input. Alternatively, you can say that every function has an implicit input parameter of the the current program state, but that turns every function on one bit of input to a potentially astronomically complex function of gigabytes of input.
I do agree with what you're saying about the value of referential transparency and the formal definition of a function as a lambda term. In fact, I work professionally as a Haskell developer and highly value this aspect of the language design; equational reasoning is a really powerful tool.
My point is that functional programming as engineering practice and language family is broader then variants of pure lambda calculi and Haskell occupies one particular place in that broad space.
