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/client_tests.rs

@@ -105,7 +105,7 @@ async fn summarize_memories_returns_empty_for_empty_input() {
 #[test]
 fn auth_request_telemetry_context_tracks_attached_auth_and_retry_phase() {
     let auth_context = AuthRequestTelemetryContext::new(
-        Some(crate::auth::AuthMode::Chatgpt),
+        Some(codex_login::AuthMode::Chatgpt),
         &crate::api_bridge::CoreAuthProvider::for_test(Some("access-token"), Some("workspace-123")),
         PendingUnauthorizedRetry::from_recovery(UnauthorizedRecoveryExecution {
             mode: "managed",