[FEATURE] Add /fork (conversation branching) support to the Cursor extension

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

The CLI supports conversation forking via `--fork-session` (and the `/fork` slash command), allowing users to branch a conversation at any point and explore a different direction while preserving the original session.

The Claude Code extension running in Cursor is missing this: there is no `/fork` in the prompt box, so the only way to branch is to scroll back and fork from an earlier message — which loses all the context accumulated since that message. #69272 requests the same capability for the VS Code extension; this issue covers the Cursor extension specifically.

### Proposed Solution

- `/fork` slash command available in the Cursor extension prompt box (alongside existing commands like `/compact`, `/usage`, etc.), forking from the **current** point of the conversation.
- The forked session appears as a new tab, preserving the full conversation history up to the fork point.

### Alternative Solutions

Currently the only workaround is to leave the extension and resume the session in the terminal with `claude --fork-session`, which drops the IDE integration.

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

1. Mid-session, Claude proposes an approach and I want to try two directions from the same context.
2. In the CLI I would `/fork` and explore one direction per session.
3. In Cursor I cannot: I either overwrite the current direction or fork from an old message and lose everything since.

### Additional Context

VS Code counterpart: #69272. Same behavior as requested there, for the Cursor extension.

Use the CLI's --fork-session and /fork behavior as the reference, then inspect the Cursor extension's prompt-box command handling and session-tab flow. The feature is done when /fork branches from the current conversation point, preserves the history, and opens the forked session in a new tab.

Rédigé par le modèle d'indexation à partir du texte de l'issue.