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/core/src/lib.rs

@@ -119,6 +119,7 @@ pub use thread_manager::ForkSnapshot;
 pub use thread_manager::NewThread;
 pub use thread_manager::StartThreadOptions;
 pub use thread_manager::ThreadManager;
+pub use thread_manager::ThreadShutdownReport;
 pub use thread_manager::build_models_manager;
 pub use thread_manager::thread_store_from_config;
 pub use web_search::web_search_action_detail;