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-selector.tsx

@@ -70,22 +70,30 @@ export function ModelSelector({
 
   // Filter Cursor models based on enabled models from global settings
   const filteredCursorModels = CURSOR_MODELS.filter((model) => {
-    // Compare model.id directly since both model.id and enabledCursorModels use full IDs with prefix
-    return enabledCursorModels.includes(model.id as any);
+    // enabledCursorModels stores CursorModelIds which may or may not have "cursor-" prefix
+    // (e.g., 'auto', 'sonnet-4.5' without prefix, but 'cursor-gpt-5.2' with prefix)
+    // CURSOR_MODELS always has the "cursor-" prefix added in model-constants.ts
+    // Check both the full ID (for GPT models) and the unprefixed version (for non-GPT models)
+    const unprefixedId = model.id.startsWith('cursor-') ? model.id.slice(7) : model.id;
+    return (
+      enabledCursorModels.includes(model.id as any) ||
+      enabledCursorModels.includes(unprefixedId as any)
+    );
   });
 
   const handleProviderChange = (provider: ModelProvider) => {
     if (provider === 'cursor' && selectedProvider !== 'cursor') {
       // Switch to Cursor's default model (from global settings)
-      onModelSelect(`${PROVIDER_PREFIXES.cursor}${cursorDefaultModel}`);
+      // cursorDefaultModel is now canonical (e.g., 'cursor-auto'), so use directly
+      onModelSelect(cursorDefaultModel);
     } else if (provider === 'codex' && selectedProvider !== 'codex') {
       // Switch to Codex's default model (use isDefault flag from dynamic models)
       const defaultModel = codexModels.find((m) => m.isDefault);
       const defaultModelId = defaultModel?.id || codexModels[0]?.id || 'codex-gpt-5.2-codex';
       onModelSelect(defaultModelId);
     } else if (provider === 'claude' && selectedProvider !== 'claude') {
-      // Switch to Claude's default model
-      onModelSelect('sonnet');
+      // Switch to Claude's default model (canonical format)
+      onModelSelect('claude-sonnet');
     }
   };