refactor(task): align continuation ids with task_id

src/tools/delegate-task/background-task.ts

@@ -10,6 +10,7 @@ import { SessionCategoryRegistry } from "../../shared/session-category-registry"
 import { QUESTION_DENIED_SESSION_PERMISSION } from "../../shared/question-denied-session-permission"
 import { setSessionFallbackChain } from "../../hooks/model-fallback/hook"
 import { stripAgentListSortPrefix } from "../../shared/agent-display-names"
+import { buildTaskMetadataBlock } from "../../features/tool-metadata-store/task-metadata-contract"
 
 function continueSessionSetup(args: {
   taskID: string
@@ -125,6 +126,8 @@ export async function executeBackgroundTask(
       description: args.description,
       run_in_background: args.run_in_background,
       command: args.command,
+      ...(sessionId ? { taskId: sessionId } : {}),
+      backgroundTaskId: task.id,
       ...(sessionId ? { sessionId } : {}),
       ...(categoryModel ? { model: { providerID: categoryModel.providerID, modelID: categoryModel.modelID } } : {}),
     }
@@ -136,7 +139,13 @@ export async function executeBackgroundTask(
     await publishToolMetadata(ctx, unstableMeta)
 
     const taskMetadataBlock = sessionId
-      ? `\n\n<task_metadata>\nsession_id: ${sessionId}\ntask_id: ${task.id}\nbackground_task_id: ${task.id}\n</task_metadata>`
+      ? `\n\n${buildTaskMetadataBlock({
+        sessionId,
+        taskId: sessionId,
+        backgroundTaskId: task.id,
+        agent: task.agent,
+        category: args.category,
+      })}`
       : ""
 
     return `Background task launched.