Handle mentioned apps via smart context (#1412)

<!-- CURSOR_SUMMARY -->
> [!NOTE]
> Send mentioned apps (names + files) to the engine through
dyad_options, omit inline other-apps prefix when engine is enabled, and
adjust e2e to snapshot request payload.
> 
> - **Engine/Backend**:
> - Pass mentioned apps to engine via
`providerOptions['dyad-engine'].dyadMentionedApps` and forward as
`dyad_options.mentioned_apps` in `llm_engine_provider`.
> - Gate inline other-apps context: only include `otherCodebasePrefix`
when `isEngineEnabled` is false.
> - Enhance `extractMentionedAppsCodebases` to return `files` alongside
`codebaseInfo`.
> - **Tests**:
> - e2e: change `mention app (with pro)` snapshot to
`snapshotServerDump("request")` and update snapshot to assert request
payload contents.
> 
> <sup>Written by [Cursor
Bugbot](https://cursor.com/dashboard?tab=bugbot) for commit
7ddddf6c16c53cd36b4c7e4ec6a57da0616d1bb0. This will update automatically
on new commits. Configure
[here](https://cursor.com/dashboard?tab=bugbot).</sup>
<!-- /CURSOR_SUMMARY -->

e2e-tests/mention_app.spec.ts

@@ -17,5 +17,5 @@ test("mention app (with pro)", async ({ po }) => {
   await po.goToAppsTab();
   await po.sendPrompt("[dump] @app:minimal-with-ai-rules hi");
 
-  await po.snapshotServerDump("all-messages");
+  await po.snapshotServerDump("request");
 });