refactor: improve AI event handling and conflict display in chat

- AI fetches events on-demand via callbacks for better efficiency
- Add conflict detection with warning display when proposing overlapping events
- Improve event search and display in chat interface
- Load full chat history for display while limiting AI context

apps/server/src/ai/utils/eventFormatter.ts

@@ -1,30 +1,4 @@
-import {
-  CalendarEvent,
-  formatDate,
-  formatTime,
-  formatDateTime,
-} from "@calchat/shared";
+import { formatDate, formatTime, formatDateTime } from "@calchat/shared";
 
-// Re-export for backwards compatibility
+// Re-export from shared package for use in toolExecutor
 export { formatDate, formatTime, formatDateTime };
-
-/**
- * Format a list of events for display in the system prompt.
- * Output is in German with date/time formatting.
- */
-export function formatExistingEvents(events: CalendarEvent[]): string {
-  if (events.length === 0) {
-    return "Keine Termine vorhanden.";
-  }
-
-  return events
-    .map((e) => {
-      const start = new Date(e.startTime);
-      const end = new Date(e.endTime);
-      const timeStr = `${formatTime(start)} - ${formatTime(end)}`;
-      const recurring = e.isRecurring ? " (wiederkehrend)" : "";
-      const desc = e.description ? ` | ${e.description}` : "";
-      return `- ${e.title} (ID: ${e.id}) | ${formatDate(start)} ${timeStr}${recurring}${desc}`;
-    })
-    .join("\n");
-}