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/tests/suite/auth_refresh.rs

@@ -4,14 +4,14 @@ use base64::Engine;
 use chrono::Duration;
 use chrono::Utc;
 use codex_app_server_protocol::AuthMode;
-use codex_core::AuthManager;
-use codex_core::auth::AuthCredentialsStoreMode;
-use codex_core::auth::AuthDotJson;
-use codex_core::auth::REFRESH_TOKEN_URL_OVERRIDE_ENV_VAR;
-use codex_core::auth::RefreshTokenError;
-use codex_core::auth::load_auth_dot_json;
-use codex_core::auth::save_auth;
 use codex_core::error::RefreshTokenFailedReason;
+use codex_login::AuthCredentialsStoreMode;
+use codex_login::AuthDotJson;
+use codex_login::AuthManager;
+use codex_login::REFRESH_TOKEN_URL_OVERRIDE_ENV_VAR;
+use codex_login::RefreshTokenError;
+use codex_login::load_auth_dot_json;
+use codex_login::save_auth;
 use codex_login::token_data::IdTokenInfo;
 use codex_login::token_data::TokenData;
 use core_test_support::skip_if_no_network;