[codex] add browser-use provider

[MagMueller]

Summary

• add a first-class browser-use provider/backend to the terminal CLI and SDK path
• route it through the Browser Use LLM Gateway chat-completions endpoint with bu-3-max as the default model
• tag Browser Use gateway requests as rust_agent while leaving existing OpenAI, Anthropic, OpenRouter, DeepSeek, Codex, and fake paths unchanged

Validation

• cargo fmt
• cargo test -p browser-use-llm build_body_merges_provider_options_without_overriding_core_fields
• cargo test -p browser-use-agent browser_use
• cargo test -p browser-use-agent provider_backend_covers_all_variants
• cargo test -p browser-use-cli cli_browser_use_backend_defaults_to_bu3_max

---

Summary by cubic

Adds a first-class browser-use provider to the CLI and SDK, routed through the Browser Use LLM Gateway with bu-3-max as the default model. Requests are tagged rust_agent, and provider options are merged into OpenAI-compatible bodies without overriding core fields.

• New Features

  • Added BrowserUse backend and provider id browser-use, defaulting to https://llm.api.browser-use.com/v1.
  • Accepts ids browser-use, browser_use, or browseruse; SDK resolves to the BrowserUse backend.
  • New CLI commands: run_browser_use and run_browser_use_session (default model bu-3-max).
  • Env-based detection: BROWSER_USE_API_KEY or LLM_BROWSER_BROWSER_USE_API_KEY; optional LLM_BROWSER_BROWSER_USE_BASE_URL.
  • Agent sets provider_options: { "request_type": "rust_agent" } and merges provider options safely into the request body.

• Migration

  • Set BROWSER_USE_API_KEY or LLM_BROWSER_BROWSER_USE_API_KEY (or run auth login browser-use-cloud).

^{Written for commit bd52a1a. Summary will update on new commits.}