improve custom installation experience / docs

Describe the problem you are trying to solve

Install Rust tooling in custom locations on my file system via env vars.

Describe the solution you'd like

I'd like rustup to not error if installed somewhere other than ~/.cargo.

Notes

$ echo $RUSTUP_HOME $CARGO_HOME $CARGO_INSTALL_ROOT
/Users/taryn/.data/rustup /Users/taryn/.data/cargo /Users/taryn/.bin

$ curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | RUSTUP_HOME="$RUSTUP_HOME" CARGO_HOME="$CARGO_HOME" sh

[..snip..]

Rustup metadata and toolchains will be installed into the Rustup
home directory, located at:

  /Users/taryn/.data/rustup

This can be modified with the RUSTUP_HOME environment variable.

The Cargo home directory located at:

  /Users/taryn/.data/cargo

This can be modified with the CARGO_HOME environment variable.

The cargo, rustc, rustup and other commands will be added to
Cargo's bin directory, located at:

  /Users/taryn/.data/cargo/bin

[..snip..]

  stable installed - rustc 1.46.0 (04488afe3 2020-08-24)


Rust is installed now. Great!

To get started you need Cargo's bin directory (/Users/taryn/.data/cargo/bin) in
your PATH
environment variable.

To configure your current shell run source /Users/taryn/.data/cargo/env

$ which rustup cargo rustc
/Users/taryn/.data/cargo/bin/rustup
/Users/taryn/.data/cargo/bin/cargo
/Users/taryn/.data/cargo/bin/rustc

$ rustup self update
error: rustup is not installed at '/Users/taryn/.cargo'

If rustup must be installed at ~/.cargo, why have the home env vars at all?
If I need to do something else that hasn't been shown here, the docs don't make that clear to me.
AFAICT I've done everything correctly but cannot use Rust tooling.

The issue names no repository files or tests. Start by reproducing the shown RUSTUP_HOME and CARGO_HOME installation, then trace the rustup self update path and existing installation documentation. Done means custom locations work consistently, or the documented setup and limitations clearly match the observed behavior.