Auto merge of #85746 - m-ou-se:io-error-other, r=joshtriplett

Redefine `ErrorKind::Other` and stop using it in std.

This implements the idea I shared yesterday in the libs meeting when we were discussing how to handle adding new `ErrorKind`s to the standard library: This redefines `Other` to be for *user defined errors only*, and changes all uses of `Other` in the standard library to a `#[doc(hidden)]` and permanently `#[unstable]` `ErrorKind` that users can not match on. This ensures that adding `ErrorKind`s at a later point in time is not a breaking change, since the user couldn't match on these errors anyway. This way, we use the `#[non_exhaustive]` property of the enum in a more effective way.

Open questions:
- How do we check this change doesn't cause too much breakage? Will a crate run help and be enough?
- How do we ensure we don't accidentally start using `Other` again in the standard library? We don't have a `pub(not crate)` or `#[deprecated(in this crate only)]`.

cc https://github.com/rust-lang/rust/pull/79965

cc `@rust-lang/libs` `@ijackson`

r? `@dtolnay`

library/std/src/io/mod.rs

@@ -1644,7 +1644,7 @@ pub trait Write {
                 if output.error.is_err() {
                     output.error
                 } else {
-                    Err(Error::new_const(ErrorKind::Other, &"formatter error"))
+                    Err(Error::new_const(ErrorKind::Uncategorized, &"formatter error"))
                 }
             }
         }