Local CLI sessions and Cloud/App sessions don't share history — feels like losing work

### What happened

I was working on a project with Claude Code from the local terminal (a long session, hours of conversation, accumulated context and decisions). At some point the app closed unexpectedly. When I reopened it and used the "work on this project remotely" option, it created a brand new, completely empty cloud session — with no indication that a local session with my full history still existed.

I only found out my previous conversation was still intact because I asked Claude directly, and it explained that local CLI sessions (`~/.claude/projects/...`) and cloud sessions are separate stores that never sync. I had to go back to a raw terminal and run `claude --resume` to recover it.

### Why this is a problem

Most users are not developers who understand session storage internals. Many people "vibe-code" through the GUI without ever touching a terminal. If their app closes and they're offered "continue in the cloud," they will reasonably assume that continues their work — not realize it silently starts from zero. This reads as data loss even when it technically isn't, and it's very frustrating to face what looks like an empty project.

### Related prior reports

- Related: #56038 (requesting sync across CLI, Desktop, and web). This issue specifically focuses on the unexpected crash / "continue in the cloud" flow that leads to perceived data loss for non-technical users.

### Suggestions

- When offering to open a cloud/remote session for a project that already has local session history, the app should detect that and either offer to resume/import that local session, or at minimum warn clearly: *"This starts a new, separate session. Your local session '' will not be included."*
- Longer term, consider letting local CLI sessions be resumable from within the desktop/web app UI directly (not just via terminal `--resume`), so users aren't forced back to the command line to avoid losing context.

Start with the “work on this project remotely” flow and the local session store at ~/.claude/projects/; reproduce the behavior after closing a CLI session and invoking that option. Compare it with claude --resume. Done means the flow either offers the existing local session or clearly warns that the cloud session is separate, with coverage for the crash-and-reopen path.

索引モデルが issue の本文から書いたものです。