feat: implement cursor model migration and enhance auto mode functionality

This commit introduces significant updates to the cursor model handling and auto mode features. The cursor model IDs have been standardized to a canonical format, ensuring backward compatibility while migrating legacy IDs. New endpoints for starting and stopping the auto mode loop have been added, allowing for better control over project-specific auto mode operations.

Key changes:
- Updated cursor model IDs to use the 'cursor-' prefix for consistency.
- Added new API endpoints: `/start` and `/stop` for managing auto mode.
- Enhanced the status endpoint to provide detailed project-specific auto mode information.
- Improved error handling and logging throughout the auto mode service.
- Migrated legacy model IDs to their canonical counterparts in various components.

This update aims to streamline the user experience and ensure a smooth transition for existing users while providing new functionalities.

apps/ui/src/components/views/board-view/shared/model-constants.ts

@@ -9,7 +9,7 @@ import { Brain, Zap, Scale, Cpu, Rocket, Sparkles } from 'lucide-react';
 import { AnthropicIcon, CursorIcon, OpenAIIcon, OpenCodeIcon } from '@/components/ui/provider-icon';
 
 export type ModelOption = {
-  id: string; // Claude models use ModelAlias, Cursor models use "cursor-{id}"
+  id: string; // All model IDs use canonical prefixed format (e.g., "claude-sonnet", "cursor-auto")
   label: string;
   description: string;
   badge?: string;
@@ -17,23 +17,27 @@ export type ModelOption = {
   hasThinking?: boolean;
 };
 
+/**
+ * Claude models with canonical prefixed IDs
+ * UI displays short labels but stores full canonical IDs
+ */
 export const CLAUDE_MODELS: ModelOption[] = [
   {
-    id: 'haiku',
+    id: 'claude-haiku', // Canonical prefixed ID
     label: 'Claude Haiku',
     description: 'Fast and efficient for simple tasks.',
     badge: 'Speed',
     provider: 'claude',
   },
   {
-    id: 'sonnet',
+    id: 'claude-sonnet', // Canonical prefixed ID
     label: 'Claude Sonnet',
     description: 'Balanced performance with strong reasoning.',
     badge: 'Balanced',
     provider: 'claude',
   },
   {
-    id: 'opus',
+    id: 'claude-opus', // Canonical prefixed ID
     label: 'Claude Opus',
     description: 'Most capable model for complex work.',
     badge: 'Premium',
@@ -43,11 +47,11 @@ export const CLAUDE_MODELS: ModelOption[] = [
 
 /**
  * Cursor models derived from CURSOR_MODEL_MAP
- * ID is prefixed with "cursor-" for ProviderFactory routing (if not already prefixed)
+ * IDs already have 'cursor-' prefix in the canonical format
  */
 export const CURSOR_MODELS: ModelOption[] = Object.entries(CURSOR_MODEL_MAP).map(
   ([id, config]) => ({
-    id: id.startsWith('cursor-') ? id : `cursor-${id}`,
+    id, // Already prefixed in canonical format
     label: config.label,
     description: config.description,
     provider: 'cursor' as ModelProvider,