Clippy can run the wrong rustc, leading to "incompatible version of rustc" errors

Summary

I had several versions of Rust installed (via rustup) and was surprised that Clippy would sometimes give errors like this:

error[E0514]: found crate `paste` compiled by an incompatible version of rustc
...
  = help: please recompile that crate using this compiler (rustc 1.64.0 ...)
  = note: the following crate versions were found:
          crate `paste` compiled by rustc 1.63.0: ...

1.64.0 was in fact the version I wanted to be using.

The surprise came when I did rustup toolchain remove 1.63.0, then cargo clean && cargo clippy, and still got the same error!

It turns out I had yet another copy of Rust 1.63.0 installed, via Homebrew, as /usr/local/bin/rustc.
I think I must have done brew install rust-analyzer at some point and that brought in the rust package.

Obviously that was bad and I just removed it. But this means cargo clippy sometimes results in two different rustc executables being used. I think that must be a bug.

Reproducer

Possible steps to reproduce, using code in https://github.com/jorendorff/incompavers:

1. Have Homebrew and rustup installed

2. Run these commands:

   brew install rust
   git clone https://github.com/jorendorff/incompavers.git
   cd incompavers
   cargo clean && cargo clippy --verbose
   

Expected behavior: Clippy should lint the crate.

Actual behavior: Clippy produced the errors below.

❯ cargo clean && cargo clippy --verbose
    Checking base64 v0.13.0
   Compiling paste v1.0.9
     Running `rustc --crate-name base64 --edition=2018 /Users/jorendorff/.cargo/registry/src/github.com-1ecc6299db9ec823/base64-0.13.0/src/lib.rs --error-format=json --json=diagnostic-rendered-ansi,artifacts,future-incompat --crate-type lib --emit=dep-info,metadata -C embed-bitcode=no -C split-debuginfo=unpacked -C debuginfo=2 --cfg 'feature="default"' --cfg 'feature="std"' -C metadata=bd37c0a895c55f89 -C extra-filename=-bd37c0a895c55f89 --out-dir /Users/jorendorff/src/incompavers/target/debug/deps -L dependency=/Users/jorendorff/src/incompavers/target/debug/deps --cap-lints allow`
     Running `rustc --crate-name paste --edition=2018 /Users/jorendorff/.cargo/registry/src/github.com-1ecc6299db9ec823/paste-1.0.9/src/lib.rs --error-format=json --json=diagnostic-rendered-ansi,artifacts,future-incompat --crate-type proc-macro --emit=dep-info,link -C prefer-dynamic -C embed-bitcode=no -C split-debuginfo=unpacked -C debuginfo=2 -C metadata=1cc8d85ad06b00de -C extra-filename=-1cc8d85ad06b00de --out-dir /Users/jorendorff/src/incompavers/target/debug/deps -L dependency=/Users/jorendorff/src/incompavers/target/debug/deps --extern proc_macro --cap-lints allow`
    Checking incompavers v0.1.0 (/Users/jorendorff/src/incompavers)
     Running `/Users/jorendorff/.rustup/toolchains/stable-x86_64-apple-darwin/bin/clippy-driver rustc --crate-name incompavers --edition=2021 src/lib.rs --error-format=json --json=diagnostic-rendered-ansi,artifacts,future-incompat --crate-type lib --emit=dep-info,metadata -C embed-bitcode=no -C split-debuginfo=unpacked -C debuginfo=2 -C metadata=8f4097126ae0e3f7 -C extra-filename=-8f4097126ae0e3f7 --out-dir /Users/jorendorff/src/incompavers/target/debug/deps -C incremental=/Users/jorendorff/src/incompavers/target/debug/incremental -L dependency=/Users/jorendorff/src/incompavers/target/debug/deps --extern base64=/Users/jorendorff/src/incompavers/target/debug/deps/libbase64-bd37c0a895c55f89.rmeta --extern paste=/Users/jorendorff/src/incompavers/target/debug/deps/libpaste-1cc8d85ad06b00de.dylib`
error[E0514]: found crate `paste` compiled by an incompatible version of rustc
 --> src/lib.rs:1:9
  |
1 | pub use paste::paste;
  |         ^^^^^
  |
  = help: please recompile that crate using this compiler (rustc 1.64.0 (a55dd71d5 2022-09-19)) (consider running `cargo clean` first)
  = note: the following crate versions were found:
          crate `paste` compiled by rustc 1.63.0: /Users/jorendorff/src/incompavers/target/debug/deps/libpaste-1cc8d85ad06b00de.dylib

error[E0514]: found crate `base64` compiled by an incompatible version of rustc
  --> src/lib.rs:22:26
   |
22 |     println!("hello {}", base64::encode(b"world"));
   |                          ^^^^^^
   |
   = help: please recompile that crate using this compiler (rustc 1.64.0 (a55dd71d5 2022-09-19)) (consider running `cargo clean` first)
   = note: the following crate versions were found:
           crate `base64` compiled by rustc 1.63.0: /Users/jorendorff/src/incompavers/target/debug/deps/libbase64-bd37c0a895c55f89.rmeta

error: could not compile `incompavers` due to 2 previous errors

Caused by:
  process didn't exit successfully: `/Users/jorendorff/.rustup/toolchains/stable-x86_64-apple-darwin/bin/clippy-driver rustc --crate-name incompavers --edition=2021 src/lib.rs --error-format=json --json=diagnostic-rendered-ansi,artifacts,future-incompat --crate-type lib --emit=dep-info,metadata -C embed-bitcode=no -C split-debuginfo=unpacked -C debuginfo=2 -C metadata=8f4097126ae0e3f7 -C extra-filename=-8f4097126ae0e3f7 --out-dir /Users/jorendorff/src/incompavers/target/debug/deps -C incremental=/Users/jorendorff/src/incompavers/target/debug/incremental -L dependency=/Users/jorendorff/src/incompavers/target/debug/deps --extern base64=/Users/jorendorff/src/incompavers/target/debug/deps/libbase64-bd37c0a895c55f89.rmeta --extern paste=/Users/jorendorff/src/incompavers/target/debug/deps/libpaste-1cc8d85ad06b00de.dylib` (exit status: 1)

Version

rustc 1.64.0 (a55dd71d5 2022-09-19)
binary: rustc
commit-hash: a55dd71d5fb0ec5a6a3a9e8c27b2127ba491ce52
commit-date: 2022-09-19
host: x86_64-apple-darwin
release: 1.64.0
LLVM version: 14.0.6

Additional Labels

No response

Start by reproducing the issue with cargo clean && cargo clippy --verbose using both rustup and Homebrew Rust installations. Inspect the reported clippy-driver, rustc, and toolchain paths to determine where the executables diverge. Done means the reproducer lints successfully without incompatible-rustc errors when the selected rustup toolchain is used.