Add codex-core public API listing (#20243)

Summary:
- Add a checked-in codex-core public API listing generated by
cargo-public-api.
- Add scripts/regen-public-api.sh with an embedded crate list,
auto-install for cargo-public-api 0.51.0, pinned nightly, and --check
mode.
- Add Rust CI jobs on the codex Linux x64 runner pool to verify the
listing stays up to date.

Testing:
- bash -n scripts/regen-public-api.sh
- just regen-public-api --check
- yq '.' .github/workflows/rust-ci.yml
.github/workflows/rust-ci-full.yml
- git diff --check

codex-rs/thread-manager-sample/README.md

@@ -1,7 +1,8 @@
 # ThreadManager Sample
 
-Small one-shot binary that starts a Codex thread with `ThreadManager`, submits a
-single user turn, and prints the final assistant message.
+Small one-shot binary that starts a Codex thread with `ThreadManager` from
+`codex-core-api`, submits a single user turn, and prints the final assistant
+message.
 
 ```sh
 cargo run -p codex-thread-manager-sample -- "Say hello"