## Why
#31323 introduces `codex-http-client` and leaves compatibility
re-exports in `codex-client`. Low-level HTTP consumers should depend on
the crate that now owns those APIs rather than continuing through the
transitional compatibility layer.
This stacked follow-up makes that ownership explicit and moves the
repository toward enforcing the abstraction without mixing call-site
churn into the extraction itself.
## What changed
- Switched `codex-backend-client`, `codex-cloud-tasks`,
`codex-exec-server`, `codex-login`, and `codex-model-provider` from
`codex-client` to `codex-http-client` where they only use low-level HTTP
APIs.
- Added the direct dependency to `codex-api` for its custom-CA request
and websocket paths while retaining `codex-client` for higher-level
retry and transport policy.
- Updated imports and normalized login's internal client type name from
`CodexHttpClient` to `HttpClient`, while preserving its existing
`CodexRequestBuilder` re-export.
- Updated `Cargo.lock` to reflect the new direct dependency edges.
## Review guide
This PR is intentionally mechanical: 20 files and 92 changed lines, with
no runtime logic changes. The largest diff is
`codex-rs/login/src/auth/default_client.rs`, where the only
semantic-looking changes are type and import renames. The remaining
source changes replace `codex_client` import paths with
`codex_http_client`; the manifest and lockfile changes mirror those
imports.
## Test plan
- Compile-checked `codex-api`, `codex-backend-client`,
`codex-cloud-tasks`, `codex-exec-server`, `codex-login`, and
`codex-model-provider` together.
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