Add complete settlement system with visual distinction

- Add settlement category with handshake emoji (🤝)
- Create settlement page for recording debt payments with user → user flow
- Implement settlement detection and visual styling across all views
- Add conditional "Settle Debts" button (hidden when balance is 0)
- Style settlement payments distinctly in recent activity with large profile pictures
- Add settlement flow styling in payments overview with green theme
- Update backend validation and Mongoose schema for settlement category
- Fix settlement receiver detection with proper user flow logic

🤝 Generated with [Claude Code](https://claude.ai/code)

Co-Authored-By: Claude <noreply@anthropic.com>

src/lib/utils/categories.ts

@@ -22,6 +22,10 @@ export const PAYMENT_CATEGORIES = {
   fun: {
     name: 'Fun',
     emoji: '🎉'
+  },
+  settlement: {
+    name: 'Settlement',
+    emoji: '🤝'
   }
 } as const;