Rename AuthMode to Backend, fix LiteLLM variant typo, add image update alerts, clean up Settings

- Fix serde deserialization error: TypeScript sent "lit_llm" but Rust expected "lite_llm"
- Rename AuthMode enum to Backend across Rust and TypeScript (with serde alias for backward compat)
- Add container image update checking via registry digest comparison
- Improve Settings page: fix image address display spacing, remove per-project auth section
- Update UI labels from "Auth" to "Backend" throughout

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

app/src/components/settings/ApiKeyInput.tsx

@@ -1,9 +1,9 @@
 export default function ApiKeyInput() {
   return (
     <div>
-      <label className="block text-sm font-medium mb-1">Authentication</label>
+      <label className="block text-sm font-medium mb-1">Backend</label>
       <p className="text-xs text-[var(--text-secondary)] mb-3">
-        Each project can use <strong>claude login</strong> (OAuth, run inside the terminal) or <strong>AWS Bedrock</strong>. Set auth mode per-project.
+        Each project can use <strong>claude login</strong> (OAuth, run inside the terminal) or <strong>AWS Bedrock</strong>. Set backend per-project.
       </p>
     </div>
   );