docs(user-guide): use brew install rustup instead of rustup-init

Author: rami3l

doc/user-guide/src/installation/other.md

@@ -38,6 +38,21 @@ $ curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh -s -- --profile
 > If you have encountered any problems installing `rustup` with a package manager,
 > please contact the package maintainer(s) for further information.
 
+In general, installing rustup with your package manager means installing a rustup binary to 
+somewhere accessible via your `$PATH`.
+To complete installing Rust, you will need to find out the name of that binary:
+
+- If the binary is named `rustup`, picking a default toolchain would usually be enough.
+  A popular choice is to set the default toolchain to `stable`:
+
+  ```sh
+  rustup default stable
+  ```
+
+- If the binary is named `rustup-init`, simply execute `rustup-init` and follow the instructions.
+  When the installation is complete, make sure that `$HOME/.cargo/bin` is in your `$PATH`,
+  and you should be able to use `rustup` normally.
+
 ### APT
 
 Starting from Debian 13 (trixie) and Ubuntu 24.04 (noble),
@@ -49,17 +64,16 @@ $ sudo apt install rustup
 
 ### Homebrew
 
-You can use `brew` to install `rustup-init`[^not-rust]:
+You can use `brew` to install `rustup`[^not-rust]:
 
 ```sh
-$ brew install rustup-init
+$ brew install rustup
 ```
 
-Then execute `rustup-init` to proceed with the installation.
-
-When the installation is complete,
-make sure that `$HOME/.cargo/bin` is in your `$PATH`,
-and you should be able to use `rustup` normally.
+Please note that Rust tools like `rustc` and `cargo` are not available via `$PATH` by default
+in this `rustup` distribution
+(see [homebrew-core#177582](https://github.com/Homebrew/homebrew-core/pull/177582) for more details).
+You might want to add `$(brew --prefix rustup)/bin` to `$PATH` to make them easier to access.
 
 [^not-rust]: This is not to be confused with the `rust` package,
 which is a `brew`-managed `rust` toolchain installation.