docs: move serialization snippet to overview.ts for type checking

Address @pgrayy's review nit: extract the inline TypeScript code block
from index.mdx into overview.ts with --8<-- markers, matching the
existing pattern used by all other code snippets on this page.

Also drops 'wire-safe' from the async-iterators.ts comment per
earlier review feedback to use 'serialization' language consistently.

Author: agent-of-mkmeral

src/content/docs/user-guide/concepts/streaming/async-iterators.ts

@@ -41,7 +41,7 @@ async function expressExample() {
     })
 
     for await (const event of agent.stream(prompt)) {
-      // Events automatically serialize to compact, wire-safe JSON via toJSON().
+      // Events automatically serialize to compact JSON via toJSON().
       // Only relevant data fields are included — the full Agent instance,
       // Tool classes, and mutable hook flags (cancel/retry) are excluded.
       res.write(`${JSON.stringify(event)}\n`)

src/content/docs/user-guide/concepts/streaming/index.mdx

@@ -228,29 +228,7 @@ Properties that hold large runtime references — such as `agent`, `orchestrator
 The same behavior applies to multi-agent and A2A events. Multi-agent events include identifying context like `nodeId` and `nodeType` alongside their data, while the `orchestrator` and `state` references are filtered out. This means you can serialize any event from `agent.stream()` or a multi-agent orchestrator's stream without additional processing:
 
 ```typescript
-for await (const event of agent.stream('Hello')) {
-  // Every event is a class instance. In-process, it carries the full agent reference:
-  //
-  //   event.type   → "modelStreamUpdateEvent"
-  //   event.agent  → LocalAgent { messages, model, tools, hooks, ... }
-  //   event.event  → { type: "modelContentBlockDeltaEvent",
-  //                     delta: { type: "textDelta", text: "Hi" } }
-
-  // JSON.stringify() calls the event's toJSON() automatically.
-  // The agent reference and other runtime objects are stripped out,
-  // leaving only the type discriminator and relevant data fields:
-  //
-  //   JSON.stringify(event) →
-  //   {
-  //     "type": "modelStreamUpdateEvent",
-  //     "event": {
-  //       "type": "modelContentBlockDeltaEvent",
-  //       "delta": { "type": "textDelta", "text": "Hi" }
-  //     }
-  //   }
-
-  res.write(`data: ${JSON.stringify(event)}\n\n`)
-}
+--8<-- "user-guide/concepts/streaming/overview.ts:event_serialization"
 ```
 </Tab>
 </Tabs>

src/content/docs/user-guide/concepts/streaming/overview.ts

@@ -120,3 +120,34 @@ async function subAgentStreamingExample() {
 
   // --8<-- [end:sub_agent_basic]
 }
+
+// Event Serialization Example
+async function eventSerializationExample() {
+  const agent = new Agent()
+
+  // --8<-- [start:event_serialization]
+  for await (const event of agent.stream('Hello')) {
+    // Every event is a class instance. In-process, it carries the full agent reference:
+    //
+    //   event.type   → "modelStreamUpdateEvent"
+    //   event.agent  → LocalAgent { messages, model, tools, hooks, ... }
+    //   event.event  → { type: "modelContentBlockDeltaEvent",
+    //                     delta: { type: "textDelta", text: "Hi" } }
+
+    // JSON.stringify() calls the event's toJSON() automatically.
+    // The agent reference and other runtime objects are stripped out,
+    // leaving only the type discriminator and relevant data fields:
+    //
+    //   JSON.stringify(event) →
+    //   {
+    //     "type": "modelStreamUpdateEvent",
+    //     "event": {
+    //       "type": "modelContentBlockDeltaEvent",
+    //       "delta": { "type": "textDelta", "text": "Hi" }
+    //     }
+    //   }
+
+    console.log(`data: ${JSON.stringify(event)}`)
+  }
+  // --8<-- [end:event_serialization]
+}