fix(#338): Log auth errors and distinguish backend down from logged out

- Add error logging for auth check failures in development mode
- Distinguish network/backend errors from normal unauthenticated state
- Expose authError state to UI (network | backend | null)
- Add comprehensive tests for error handling scenarios

Refs #338

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

apps/web/src/app/(auth)/callback/page.test.tsx

@@ -33,6 +33,7 @@ describe("CallbackPage", (): void => {
       user: null,
       isLoading: false,
       isAuthenticated: false,
+      authError: null,
       signOut: vi.fn(),
     });
   });
@@ -49,6 +50,7 @@ describe("CallbackPage", (): void => {
       user: null,
       isLoading: false,
       isAuthenticated: false,
+      authError: null,
       signOut: vi.fn(),
     });
 
@@ -138,6 +140,7 @@ describe("CallbackPage", (): void => {
       user: null,
       isLoading: false,
       isAuthenticated: false,
+      authError: null,
       signOut: vi.fn(),
     });