My point is that the definition of functional programming is this purity you mention not using first-class functions. Pure means that something has more of that thing you want and impure means it has less - so pure functional programming has more 'functional programming' then 'standard functional programming'. Sure completely pure functional programming is impractical - there are degrees to how purely functional are the languages we use in practice - but this is the point practical languages are more or less functional and how much are they functional depends on how much they depend on state manipulation not on using higher order functions or having first class functions.

Defining functional programming as programming with first-class functions is a consistent definition and could be used if not the fact that more people use the other definition (at least that is what the wikipedia definition and other online material makes me believe).