fix(stream): TEST-tab chat — resolve liveChatId from snippet, poll until the broadcast is live

The open creator report ('I still cannot post a chat message in the TEST tab',
feedback 'Chat polling couldn't start — Mock Chat Input is disabled') was a single
cause: the liveChatId never resolved. Two YouTube API facts:
1. The id lives in snippet.liveChatId — contentDetails has no such property
   (TASK 44 read part=contentDetails: could never resolve).
   https://developers.google.com/youtube/v3/live/docs/liveBroadcasts
2. It only exists once the broadcast is LIVE — the official sample lists
   broadcastStatus=active, and our fetch ran before the frame pump pushed RTMP
   (enableAutoStart flips ready→live). A ready-state list legitimately returns
   no id.
   https://github.com/youtube/api-samples/blob/master/java/src/main/java/com/google/api/services/samples/youtube/cmdline/live/GetLiveChatId.java

Fix: GetBroadcastLiveChatIdAsync reads part=snippet and polls with a bounded
retry (10x/2s); PrepareAndStartLiveAsync starts the frame pump FIRST, then
resolves the id. Chat stays non-fatal. Docs ai.md/TASKS.md/HANDOFF.md +
MyMistakes recipe updated same commit. TASK 44.

Good Dog: ONE integration test (GetBroadcastLiveChatIdAsync_Polls_Snippet_...
) drives a broadcast that gains its id mid-retry and asserts every request used
part=snippet. Gate: clean build 0 warnings, vstest 318/318.

ai.md

@@ -1560,6 +1560,15 @@ These are the hard facts behind every decision. Full list in `TASKS.md`.
   `StopStream` clears it in every path (including pump-failure rollback). Once the liveChatId
   resolves, `TestSession.BeginTest(broadcastId, liveChatId)` opens the **TEST drawer** (third
   right-rail pull-out, three-way one-open-at-a-time exclusivity wired in `MainViewModel.Ypp.cs`).
+  **liveChatId resolution (fixed 2026-09-25, TASK 44):** the id is `snippet.liveChatId` —
+  `contentDetails` has NO liveChatId property — and YouTube only populates it once the broadcast
+  is LIVE (the official GetLiveChatId.java sample lists broadcastStatus=active). So
+  `GetBroadcastLiveChatIdAsync(broadcastId, maxAttempts=10, delayMs=2000)` reads
+  part=snippet and polls with a bounded retry, and `PrepareAndStartLiveAsync` starts the frame
+  pump FIRST (the RTMP push is what flips ready→live via enableAutoStart) before resolving it.
+  Chrome-debugging note: a fetch at broadcast-insert time (lifecycleStatus ready) legitimately
+  returns no id — missing liveChatId is the ready state, not an auth failure, until ~20s post
+  push.
   Drawer tooling: **Mock Chat Input** = a REAL `liveChat/messages.insert` (`YouTubeStreamService.InsertChatMessageAsync`)
   that round-trips through the real ~2s poll and renders through the live overlay path;
   **simulated Subscriber / New Member / Super Chat** = `YouTubeChatService.InjectSimulatedMessage`,