fix(copilot): mark internal hook injections as agent-initiated

Apply the internal initiator marker to automated continuation, recovery, babysitter, stop-hook, and hook-message injections so Copilot attribution consistently sets x-initiator=agent for system-generated prompts.

src/hooks/unstable-agent-babysitter/index.test.ts

@@ -1,6 +1,7 @@
 import { afterEach, describe, expect, test } from "bun:test"
 import { _resetForTesting, setMainSession } from "../../features/claude-code-session-state"
 import type { BackgroundTask } from "../../features/background-agent"
+import { OMO_INTERNAL_INITIATOR_MARKER } from "../../shared/internal-initiator-marker"
 import { createUnstableAgentBabysitterHook } from "./index"
 
 const projectDir = process.cwd()
@@ -93,6 +94,7 @@ describe("unstable-agent-babysitter hook", () => {
     expect(text).toContain("background_output")
     expect(text).toContain("background_cancel")
     expect(text).toContain("deep thought")
+    expect(text).toContain(OMO_INTERNAL_INITIATOR_MARKER)
   })
 
   test("fires reminder for hung minimax task", async () => {
@@ -128,6 +130,7 @@ describe("unstable-agent-babysitter hook", () => {
     expect(text).toContain("background_output")
     expect(text).toContain("background_cancel")
     expect(text).toContain("minimax thought")
+    expect(text).toContain(OMO_INTERNAL_INITIATOR_MARKER)
   })
 
   test("does not remind stable model tasks", async () => {

src/hooks/unstable-agent-babysitter/unstable-agent-babysitter-hook.ts

@@ -1,7 +1,7 @@
 import type { BackgroundManager } from "../../features/background-agent"
 import { getMainSessionID, getSessionAgent } from "../../features/claude-code-session-state"
 import { log } from "../../shared/logger"
-import { resolveInheritedPromptTools } from "../../shared"
+import { createInternalAgentTextPart, resolveInheritedPromptTools } from "../../shared"
 import {
   buildReminder,
   extractMessages,
@@ -158,7 +158,7 @@ export function createUnstableAgentBabysitterHook(ctx: BabysitterContext, option
             ...(agent ? { agent } : {}),
             ...(model ? { model } : {}),
             ...(tools ? { tools } : {}),
-            parts: [{ type: "text", text: reminder }],
+            parts: [createInternalAgentTextPart(reminder)],
           },
           query: { directory: ctx.directory },
         })