BranchPickerWidget uses sessions-specific CSS class names in workbench core

The `BranchPickerWidget` in `src/vs/workbench/contrib/chat/browser/widget/input/branchPickerWidget.ts` uses `.sessions-chat-picker-slot` and `.sessions-chat-dropdown-label` CSS class names. These styles are defined only in `src/vs/sessions/contrib/chat/browser/media/chatWidget.css`, so non-sessions consumers would lack styling.

**Proposed fix:**
- Rename `.sessions-chat-picker-slot` → `.chat-picker-slot` and `.sessions-chat-dropdown-label` → `.chat-dropdown-label` across all ~8 picker TS files and ~4 CSS files
- Extract the base picker-slot styles (~120 lines, chatWidget.css lines 312-436) into a new CSS file next to `branchPickerWidget.ts` in the workbench layer
- Keep sessions-specific overrides (responsive breakpoints, workspace/session-type sizing) in sessions CSS with the updated class names

**Context:** PR #325712 migrated the BranchPickerWidget to core. The class names were kept as-is to avoid a noisy cross-cutting rename in the same PR.

ref #325712