docs(chat): describe the action-turn branch, and fix two Head Start tool statements

The actions lifecycle flow only covered the edit-only path; it now says what
happens when onAction returns chat.turn(). The Head Start step said the bound
streamText already carried the caller's tools, and the migration example kept
a spread comment for a call with no spread.

Author: ericallam

docs/ai-chat/actions.mdx

@@ -52,6 +52,8 @@ export const myChat = chat.agent({
 
 **Lifecycle flow:** Wake → parse action against `actionSchema` → `hydrateMessages` (if set) → **`onAction`** → apply `chat.history` mutations → emit `trigger:turn-complete` → wait for next message.
 
+When `onAction` returns `chat.turn()`, the flow continues instead of emitting `trigger:turn-complete`: the edit is snapshotted, then a turn runs on the edited history with `trigger: "action-turn"`, so `onTurnStart`, `run()`, `onBeforeTurnComplete` and `onTurnComplete` all fire and the answer is persisted like any turn's. See [Answering after an action](#answering-after-an-action).
+
 ## Answering after an action
 
 An action is a state edit. To answer after the edit, return `chat.turn()`: the edit is applied and snapshotted, then a turn runs on the edited history exactly as a message turn does. `onTurnStart`, `run()`, `onBeforeTurnComplete` and `onTurnComplete` fire, the turn counter advances, and the answer gets everything a turn has: the agent's system prompt and tools, steering, compaction, injected instructions and persistence.

docs/ai-chat/fast-starts.mdx

@@ -251,7 +251,7 @@ This is an **import-chain** problem, not a runtime one. A "we'll strip the execu
     ```
   </Step>
   <Step title="Build the head-start handler">
-    Call `chat.headStart({ agentId, run })`. It returns a standard Web Fetch handler: `(req: Request) => Promise<Response>`. The `run` callback receives a `streamText` that already carries the SDK-owned wiring: the converted messages, your schema-only tools, `stopWhen: stepCountIs(1)` and the abort signal. Add your own `model` and `system` on top.
+    Call `chat.headStart({ agentId, run })`. It returns a standard Web Fetch handler: `(req: Request) => Promise<Response>`. The `run` callback receives a `streamText` that already carries the SDK-owned wiring: the converted messages, `stopWhen: stepCountIs(1)` and the abort signal. Pass your schema-only tools to it explicitly, and add your own `model` and `system` on top.
 
     ```ts lib/chat-handler.ts
     import { chat } from "@trigger.dev/sdk/chat-server";

docs/ai-chat/migrating-from-a-route-handler.mdx

@@ -147,7 +147,6 @@ export const myChat = chat.agent({
   tools,
   run: async ({ messages, tools, signal, streamText }) =>
     streamText({
-      // Spread first, so every option below still wins.
       tools,
       model: anthropic("claude-sonnet-4-5"),
       system: "You are a helpful assistant.",