Fix Claude backend CLI compatibility

[HansBug]

Summary

Fixes the optional Claude backend against current Claude Code CLI behavior:

• add --verbose when using claude -p --output-format stream-json;
• run backend subprocesses with the requested --cwd as their actual process cwd, while keeping the existing prompt instruction behavior;
• resume Claude threads with --resume <sessionId> instead of Codex's positional resume <threadId> syntax;
• write --save-response from parsed assistant text for non-Codex backends, since Claude does not support Codex's --output-last-message flag.

This keeps Codex backend args unchanged and avoids introducing Claude-only flags beyond the currently required --verbose.

Fixes #3.

Testing

• npm run typecheck
• npm run lint
• npm run build
• npm run test

Note: I did not include a full-repo npm run format change because the current main branch already reports Prettier warnings across unrelated files. The files touched by this PR were formatted where needed.