Rust's closures do not heap allocate unless you box them, like any other struct, because closures are sugar for a struct + a function, that's correct.

(and yes, there are three types of closures, because they need to know if they take said struct by reference, by mutable reference, or by owner.)

It does have to, because of the way mutation and ownership work. Which is great! But it makes functional programming awkward. The language does not "steward" you toward function composition.