feat: label AI-rescheduled appointments and add manual confirm option

- Add movedByAi boolean column to Appointment table (default false)
- reschedule-graph: set movedByAi=true when AI moves an appointment
- PATCH /api/appointments/:id/confirm endpoint clears the movedByAi flag
- Schedule grid: show teal AI badge on appointment cards where movedByAi=true
- Right-click context menu: 'Manually Confirmed' removes the AI badge via the confirm endpoint

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

apps/Backend/src/ai/reschedule-graph.ts

@@ -439,6 +439,7 @@ async function moveAppointment(
     startTime: newStartTime,
     endTime:   newEndTime,
     status:    "scheduled",
+    movedByAi: true,
   } as any);
 
   return "ok";

apps/Backend/src/routes/appointments.ts

@@ -417,6 +417,19 @@ router.put(
   }
 );
 
+// Manually confirm an AI-moved appointment (clears the movedByAi flag)
+router.patch("/:id/confirm", async (req: Request, res: Response): Promise<any> => {
+  try {
+    const id = parseInt(req.params.id);
+    if (isNaN(id)) return res.status(400).json({ message: "Invalid appointment ID" });
+    const updated = await storage.updateAppointment(id, { movedByAi: false } as any);
+    return res.json(updated);
+  } catch (err) {
+    const msg = err instanceof Error ? err.message : String(err);
+    return res.status(500).json({ message: msg });
+  }
+});
+
 // Delete an appointment
 router.delete("/:id", async (req: Request, res: Response): Promise<any> => {
   try {