refactor(coding): organize product internals

[ktsoator]

Summary

• move the browser client from web to coding/client and keep Coding product implementation under coding/internal
• establish app -> httpapi -> conversation -> engine ownership boundaries without depending on harness
• keep process configuration separate from persisted provider and conversation settings
• split conversation lifecycle and provider storage by responsibility, with lifecycle and rollback coverage

Verification

• go test ./...
• go test -race ./coding/...
• git diff --check
• bun run build in coding/client
• bun run lint in coding/client (passes with the existing App.tsx:171 exhaustive-deps warning)