Add Codex Desktop spoof toggle and wire provider start options end-to-end

[rexlManu]

Summary

• Adds a new app setting, Spoof Codex Desktop client, to opt into Codex Desktop-style session startup metadata.
• Extends Codex provider start options with spoofAsCodexDesktop and threads provider options from web composer commands through orchestration to server session startup.
• Updates Codex app-server initialization to send Desktop client metadata/capabilities and set serviceName: "codex_desktop" when spoofing is enabled.
• Extracts provider start option schemas into packages/contracts/src/providerOptions.ts and re-exports them for shared contract usage.
• Adds/updates tests across contracts, server orchestration/adapter/manager, and web app settings to validate schema acceptance and option forwarding.
• Includes small reliability/type-safety fixes in web sync/chat tests (status: "starting", safer timer/resolve typing, welcome sync flush path).

Testing

• Not run (in this PR write-up context).
• Added contract coverage for thread.turn.start and provider session start input parsing with providerOptions.codex.spoofAsCodexDesktop.
• Added server coverage to verify provider options propagation through ProviderCommandReactor and CodexAdapter into session start payloads.
• Added manager coverage to verify buildCodexInitializeParams({ spoofAsCodexDesktop: true }) emits Codex Desktop metadata/capabilities.
• Added web settings coverage to verify provider options are omitted when unset and correctly derived when binary/home path + spoof toggle are enabled.