core: remove cross-crate re-exports from lib.rs (#16512)

## Why

`codex-core` was re-exporting APIs owned by sibling `codex-*` crates,
which made downstream crates depend on `codex-core` as a proxy module
instead of the actual owner crate.

Removing those forwards makes crate boundaries explicit and lets leaf
crates drop unnecessary `codex-core` dependencies. In this PR, this
reduces the dependency on `codex-core` to `codex-login` in the following
files:

```
codex-rs/backend-client/Cargo.toml
codex-rs/mcp-server/tests/common/Cargo.toml
```

## What

- Remove `codex-rs/core/src/lib.rs` re-exports for symbols owned by
`codex-login`, `codex-mcp`, `codex-rollout`, `codex-analytics`,
`codex-protocol`, `codex-shell-command`, `codex-sandboxing`,
`codex-tools`, and `codex-utils-path`.
- Delete the `default_client` forwarding shim in `codex-rs/core`.
- Update in-crate and downstream callsites to import directly from the
owning `codex-*` crate.
- Add direct Cargo dependencies where callsites now target the owner
crate, and remove `codex-core` from `codex-rs/backend-client`.

codex-rs/core/src/models_manager/manager.rs

@@ -1,13 +1,9 @@
 use super::cache::ModelsCacheManager;
 use crate::api_bridge::auth_provider_from_auth;
 use crate::api_bridge::map_api_error;
-use crate::auth::AuthManager;
-use crate::auth::AuthMode;
-use crate::auth::CodexAuth;
 use crate::auth_env_telemetry::AuthEnvTelemetry;
 use crate::auth_env_telemetry::collect_auth_env_telemetry;
 use crate::config::Config;
-use crate::default_client::build_reqwest_client;
 use crate::error::CodexErr;
 use crate::error::Result as CoreResult;
 use crate::model_provider_info::ModelProviderInfo;
@@ -23,6 +19,10 @@ use codex_api::ModelsClient;
 use codex_api::RequestTelemetry;
 use codex_api::ReqwestTransport;
 use codex_api::TransportError;
+use codex_login::AuthManager;
+use codex_login::AuthMode;
+use codex_login::CodexAuth;
+use codex_login::default_client::build_reqwest_client;
 use codex_otel::TelemetryAuthMode;
 use codex_protocol::config_types::CollaborationModeMask;
 use codex_protocol::openai_models::ModelInfo;

codex-rs/core/src/models_manager/manager_tests.rs

@@ -1,13 +1,13 @@
 use super::*;
-use crate::AuthManager;
-use crate::CodexAuth;
-use crate::ModelProviderAuthInfo;
-use crate::auth::AuthCredentialsStoreMode;
 use crate::config::ConfigBuilder;
 use crate::model_provider_info::WireApi;
 use base64::Engine as _;
 use chrono::Utc;
 use codex_api::TransportError;
+use codex_login::AuthCredentialsStoreMode;
+use codex_login::AuthManager;
+use codex_login::CodexAuth;
+use codex_protocol::config_types::ModelProviderAuthInfo;
 use codex_protocol::openai_models::ModelsResponse;
 use core_test_support::responses::mount_models_once;
 use http::HeaderMap;