fix: language switcher stuck in French on store dashboard

Three compounding bugs prevented language switching on the store dashboard:
- Cookie missing path="/", scoping it to the API endpoint path only
- STORE frontend resolution chain ignored the cookie entirely
- Store header used inline x-data with wrong language names instead of shared languageSelector()

Also updates architecture doc with correct per-frontend resolution priorities,
cookie name, API endpoint path, and file references.

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

middleware/language.py

@@ -73,6 +73,7 @@ class LanguageMiddleware(BaseHTTPMiddleware):
             language = resolve_store_dashboard_language(
                 user_preferred=user_preferred,
                 store_dashboard=store_dashboard,
+                cookie_language=cookie_language,
             )
 
         elif frontend_type == FrontendType.STOREFRONT:
@@ -170,6 +171,7 @@ def set_language_cookie(response: Response, language: str) -> Response:
             max_age=60 * 60 * 24 * 365,  # 1 year
             httponly=False,  # Accessible to JavaScript
             samesite="lax",
+            path="/",
         )
     return response
 
@@ -184,5 +186,5 @@ def delete_language_cookie(response: Response) -> Response:
     Returns:
         Modified response with cookie deleted
     """
-    response.delete_cookie(key=LANGUAGE_COOKIE_NAME)
+    response.delete_cookie(key=LANGUAGE_COOKIE_NAME, path="/")
     return response