docs(ai-chat): atomic onTurnComplete writes + Anthropic prose (#3693)

## Summary

Three post-merge fixes for the AI Agents docs (#3226), all caught by
review after merge.

## Fixes

- **`onTurnComplete` examples now use `db.$transaction`** — both the
Database persistence "Complete example" and the Lifecycle hooks
reference example were doing two separate `await` calls
(`db.chat.update` then `db.chatSession.upsert`). That's the exact
non-atomic pattern the warning earlier on the persistence page calls out
as ❌: a refresh between the two writes reads a stale `lastEventId` and
duplicates the assistant message on resume. Both examples now use the
recommended atomic form.

- **Background injection self-review prose aligned with the code** — the
prose said "gpt-4o-mini" but the example above it had been swapped to
`claude-haiku-4-5`. The Anthropic-sweep script only touched code blocks;
this prose line wasn't picked up.

## Test plan

- [x] Both updated examples use `db.$transaction([...])`
- [x] Prose matches the model used in the code block
- [ ] Mintlify deployment passes

docs/ai-chat/background-injection.mdx

@@ -154,7 +154,7 @@ export const myChat = chat.agent({
 });
 ```
 
-The self-review runs on `gpt-4o-mini` (fast, cheap) in the background. If the user sends another message before it completes, the coaching is still injected — `chat.inject()` persists across the idle wait.
+The self-review runs on `claude-haiku-4-5` (fast, cheap) in the background. If the user sends another message before it completes, the coaching is still injected — `chat.inject()` persists across the idle wait.
 
 ## Other use cases
 

docs/ai-chat/lifecycle-hooks.mdx

@@ -412,15 +412,18 @@ Fires after each turn completes, after the response is captured and the stream i
 export const myChat = chat.agent({
   id: "my-chat",
   onTurnComplete: async ({ chatId, uiMessages, runId, chatAccessToken, lastEventId }) => {
-    await db.chat.update({
-      where: { id: chatId },
-      data: { messages: uiMessages },
-    });
-    await db.chatSession.upsert({
-      where: { id: chatId },
-      create: { id: chatId, runId, publicAccessToken: chatAccessToken, lastEventId },
-      update: { runId, publicAccessToken: chatAccessToken, lastEventId },
-    });
+    // Atomic write — see Database persistence for the race-condition rationale
+    await db.$transaction([
+      db.chat.update({
+        where: { id: chatId },
+        data: { messages: uiMessages },
+      }),
+      db.chatSession.upsert({
+        where: { id: chatId },
+        create: { id: chatId, runId, publicAccessToken: chatAccessToken, lastEventId },
+        update: { runId, publicAccessToken: chatAccessToken, lastEventId },
+      }),
+    ]);
   },
   run: async ({ messages, signal }) => {
     return streamText({ model: anthropic("claude-sonnet-4-5"), messages, abortSignal: signal });

docs/ai-chat/patterns/database-persistence.mdx

@@ -258,16 +258,19 @@ export const myChat = chat.agent({
     });
   },
   onTurnComplete: async ({ chatId, uiMessages, runId, chatAccessToken, lastEventId }) => {
-    // Persist assistant response + stream position
-    await db.chat.update({
-      where: { id: chatId },
-      data: { messages: uiMessages },
-    });
-    await db.chatSession.upsert({
-      where: { id: chatId },
-      create: { id: chatId, runId, publicAccessToken: chatAccessToken, lastEventId },
-      update: { runId, publicAccessToken: chatAccessToken, lastEventId },
-    });
+    // Persist assistant response + stream position atomically — see the
+    // race-condition warning earlier on this page.
+    await db.$transaction([
+      db.chat.update({
+        where: { id: chatId },
+        data: { messages: uiMessages },
+      }),
+      db.chatSession.upsert({
+        where: { id: chatId },
+        create: { id: chatId, runId, publicAccessToken: chatAccessToken, lastEventId },
+        update: { runId, publicAccessToken: chatAccessToken, lastEventId },
+      }),
+    ]);
   },
   run: async ({ messages, signal }) => {
     return streamText({