Remote Control session renames do not sync to Mac Remote Control client or web interface

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

When a Remote Control session is renamed on the origin machine (Windows PC running Claude Code Desktop), the rename does not propagate to:
1. The Mac accessing the session via Remote Control in Claude Code Desktop
2. 2. The web interface at claude.ai/code
The session retains its original name on the Mac and web, while the PC displays the new name.

Example:
- Session initially created with name: "new game developer onboarding"
- - Session renamed on PC to: "l'intendant"
- - - PC shows: "l'intendant" ✅
- - - - Mac (Remote Control) shows: "new game developer onboarding" ❌
- - - - - Web (claude.ai/code) shows: "new game developer onboarding" ❌
This causes confusion about which session is being worked on across devices.

### What Should Happen?

When a Remote Control session is renamed on the Windows PC origin machine, the new name should immediately propagate to:
1. The Mac accessing the session via Remote Control in Claude Code Desktop
2. 2. The web interface at claude.ai/code
All three access points should display the same, current session name so users can easily identify which session they're working on, regardless of where they access it from.

### Error Messages/Logs

```shell

```

### Steps to Reproduce

a1. On Windows PC: Start a Claude Code Remote Control session (e.g., `/remote-control "new game developer onboarding"`)
2. Connect to that session from Mac via Claude Code Desktop app using Remote Control
3. 3. Verify the session appears with the initial name "new game developer onboarding" on the Mac - it should sync correctly ✅
4. 4. On the Windows PC, rename the session to a new name (e.g., "l'intendant") using the rename feature
5. 5. Go back to the Mac Remote Control connection - the session name should update to "l'intendant" but it doesn't ❌
6. 6. Check claude.ai/code web interface - it also still shows the old name ❌
7. 7. Verify that the Windows PC shows the new name "l'intendant" ✅ (but Mac and web don't sync it)

### Claude Model

None

### Is this a regression?

No, this never worked

### Last Working Version

_No response_

### Claude Code Version

Claude Desktop (Latest)

### Platform

Anthropic API

### Operating System

macOS

### Terminal/Shell

Terminal.app (macOS)

### Additional Information

_No response_

Start by tracing the `/remote-control` session rename flow from the Windows origin through the Mac Claude Code Desktop client and the claude.ai/code web interface. Compare how the initial session name propagates with how a later rename is handled; done means a rename appears consistently on the origin, Mac, and web clients.

Written by the indexing model from the issue text.