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;
 

src/lib/utils/settlements.ts

@@ -0,0 +1,63 @@
+// Utility functions for identifying and handling settlement payments
+
+/**
+ * Identifies if a payment is a settlement payment based on category
+ */
+export function isSettlementPayment(payment: any): boolean {
+  if (!payment) return false;
+  
+  // Check if category is settlement
+  return payment.category === 'settlement';
+}
+
+/**
+ * Gets the settlement icon for settlement payments
+ */
+export function getSettlementIcon(): string {
+  return '🤝'; // Handshake emoji for settlements
+}
+
+/**
+ * Gets appropriate styling classes for settlement payments
+ */
+export function getSettlementClasses(payment: any): string[] {
+  if (!isSettlementPayment(payment)) {
+    return [];
+  }
+  
+  return ['settlement-payment'];
+}
+
+/**
+ * Gets settlement-specific display text
+ */
+export function getSettlementDisplayText(payment: any): string {
+  if (!isSettlementPayment(payment)) {
+    return '';
+  }
+  
+  return 'Settlement';
+}
+
+/**
+ * Gets the other user in a settlement (the one who didn't pay)
+ */
+export function getSettlementReceiver(payment: any): string {
+  if (!isSettlementPayment(payment) || !payment.splits) {
+    return '';
+  }
+  
+  // Find the user who has a positive amount (the receiver)
+  const receiver = payment.splits.find(split => split.amount > 0);
+  if (receiver && receiver.username) {
+    return receiver.username;
+  }
+  
+  // Fallback: find the user who is not the payer
+  const otherUser = payment.splits.find(split => split.username !== payment.paidBy);
+  if (otherUser && otherUser.username) {
+    return otherUser.username;
+  }
+  
+  return '';
+}