They might not do that in Rust, they do in Haskell, JavaScript and plenty of others.
A "function", in the academic literature, is a form that has two parts: a set of variables that it binds (called the parameters) and a bit of encapsulated functionality (called the body). The function itself has no knowledge of the outside world. This is why the notion of free variables is important, or even relevant. A free variable is any variable that occurs unbound within the body of a function. The function doesn't know anything about the free variables; they simply exist within it.
A "closure" is a pairing of a function with an environment. If you are talking about functions and "their" environments, you are talking about closures. In most languages, a function is only syntactically valid if all of its free variables are bound within the surrounding environment. (However, if you were to implement a language with dynamic scope instead of the now-standard lexical scope, you wouldn't even check such a thing statically.)
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I'd like to try to make my point by drawing an analogy to another pair of PL terms that are often similarly conflated: "parameters" and "arguments".
A parameter is a variable that occurs in the binding context of a function definition, and is therefore considered to be bound within the body of the function. Parameters are not free variables.
An argument is a value that is passed to a function during a function call. Arguments are bound to parameters during the set-up of the function call.
Let's imagine I want to talk about an anonymous function `(λ (x) (add1 x))`. This is a function that has one parameter, `x`. It then returns the result of incrementing the value of `x` by 1 (we are assuming the value is a number).
If I had instead said that this function "takes one argument `x`", it would technically have been incorrect. When we're talking about the function, we know nothing about the arguments it will eventually take; we only have information about the parameters.
It is exceedingly common for people to use these terms interchangeably, but they are actually distinct in the academic literature, as they each refer to entirely separate (though related) things. The function/closure distinction is similar. Many people use the terms interchangeably, but they are actually meant to be separate. You are conflating them, and my comments here have been meant to educate people about the actual distinction.
The reason I feel this stuff is important to mention is because it is not uncommon to find discussions where the conflation does cause people headaches. The problem is exacerbated by the prevalence of people who claim that the difference is "irrelevant", up until the point that the difference actually matters, at which point those people are simply not around. There are a ton of small sets of terms of this nature that, when speaking to laypeople, you would think are just synonyms, except that it turns out that sometimes the technical distinction is important. It means people who actively seek out spaces to educate those who want to learn (like myself) have to do extra work to undo the faulty learning and then start over with correct definitions. Like you said: for most people it's not a big enough deal to worry about, but the distinction is present and there are times when it does matter.
I like informing people of things, and I happen to find interest in minute details, definitions of terms, and so on. I know that's not for everyone. To solve this "problem" (if I can call it that), I try to start my discussions by being honest about my intentions: I explicitly state that the point is minor or pedantic or something to that effect. That way I get to write about the things I want to write about while also giving people plenty of room to just choose not to worry about it if they don't care, and everybody comes out the other side happy. Except you, I guess. Sorry about that.
