[Bug]: Playground: renaming the project loses all changes

### Problem description

In the Playground, renaming a project seems to silently discard changes made in the editor :(

### Steps to reproduce

1. Create a playground project
2. Spend time writing code
3. Decide on a name
4. Click the rename button
5. Write the new name
6. Hit Enter
7. Oops, we're back to hello world!

### Expected behavior

At least a warning about the changes being lost. Or the changes being saved.

### Environment

_No response_

### Possible solution

_No response_

### Additional context

_No response_

Reproduce the issue in the Playground by creating a project, editing its code, and renaming it with the rename button. Trace the rename flow from the editor state and verify that completion either preserves the changes or warns before they are lost.

Written by the indexing model from the issue text.