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/plugins.rs

@@ -6,8 +6,8 @@ use std::time::Duration;
 use std::time::Instant;
 
 use anyhow::Result;
-use codex_core::CodexAuth;
 use codex_features::Feature;
+use codex_login::CodexAuth;
 use codex_protocol::protocol::EventMsg;
 use codex_protocol::protocol::Op;
 use core_test_support::apps_test_server::AppsTestServer;
@@ -400,7 +400,7 @@ async fn explicit_plugin_mentions_track_plugin_used_analytics() -> Result<()> {
     );
     assert_eq!(
         event["event_params"]["product_client_id"],
-        serde_json::json!(codex_core::default_client::originator().value)
+        serde_json::json!(codex_login::default_client::originator().value)
     );
     assert_eq!(event["event_params"]["model_slug"], "gpt-5");
     assert!(event["event_params"]["thread_id"].as_str().is_some());