The latter are important examples which usually also exist in "purely" functional programming languages.

So every program starts in the IO monad. From here you can get any input you need, call into relatively "pure" code with the inputs, and then output the result in some way. The reverse does not work. You cannot call into IO code from pure code like you can call into a Maybe function from pure code.

Purely functional programming may also be defined by forbidding state changes and mutable data.

purely functional programming usually designates a programming paradigm—a style of building the structure and elements of computer programs—that treats all computation as the evaluation of mathematical functions.

Purely functional data structures are persistent. Persistency is required for functional programming; without it, the same computation could return different results.