Multi-root workspace UX: show active CWD and make root switching obvious

Problem

When using the Claude Code extension in a multi-root workspace (VS Code / Cursor), two related UX issues make it hard to work efficiently:

1. The current working directory isn't visible in the extension UI. There's no header indicator showing which workspace root the active session is operating on. The only hint is a small chip near the chat input, which is easy to miss.
2. Switching between workspace roots is cumbersome. New sessions auto-pick the first root in the workspace. There's no quick "switch root" affordance comparable to opening a new terminal in a specific folder. Users have to fall back to slash commands (/cwd, /add-dir) or the terminal.

In contrast, the integrated terminal makes this trivial: open a new terminal, pick the folder from a dropdown, done.

Suggested improvements

• Show the active CWD prominently in the session header (not just in a small chip).
• Add a root picker next to the + (new session) button — similar to how the terminal lets you pick a folder when creating a new instance.
• Persist a per-workspace default root setting (e.g., claude-code.defaultWorkingDirectory) so the extension doesn't always grab the first root.
• Right-click → "Start Claude Code here" in the Explorer (works in VS Code, missing in Cursor as far as I can tell).

Environment

• Editor: Cursor (also reproduces in VS Code)
• OS: macOS
• Workspace type: multi-root .code-workspace