OK - I can agree with the interpretation that the 'impurity' is not inside map - but rather in the argument it takes. You can reconcile it with the mathematical definition of function by saying that some 'programming functions' take the implicit 'state' argument and that map is not one of them.

It still does not resolve the definition problem - is functional programming programming with pure mathematical functions - or is it using first-class functions (not in mathematical sense)? Which definition is more useful? Which is more widely used?