fix(team): identify the coordinator as the root agent

packages/team/tool-agent-team/README.md

@@ -128,7 +128,7 @@ A shared task board is available in this session. Task owners are agent ids from
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.
 ```

packages/team/tool-agent-team/README.zh.md

@@ -128,7 +128,7 @@ A shared task board is available in this session. Task owners are agent ids from
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.
 ```

packages/team/tool-agent-team/src/index.ts

@@ -28,7 +28,7 @@ const TEAM_POLICY = `A shared task board is available in this session. Task owne
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.`
 

snapshots/sdk/team-session-mode/system-prompt.expected.md

@@ -20,7 +20,7 @@ A shared task board is available in this session. Task owners are agent ids from
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.
 

snapshots/session/team-session-mode/system-prompt.expected.md

@@ -26,7 +26,7 @@ A shared task board is available in this session. Task owners are agent ids from
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.
 

snapshots/session/team-targets/system-prompt.expected.md

@@ -9,7 +9,7 @@ A shared task board is available in this session. Task owners are agent ids from
 
 Split write work into disjoint scopes, record expected write scopes on shared tasks, and use task dependencies when work must be ordered. Write-scope overlap is advisory, not a lock.
 
-Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The Lead must wait for required teammates before giving the final answer.
+Shared-task workflow is list, get, claim with the current revision, perform the work, then complete. Task readiness never starts an owner. The root agent must wait for required teammates before giving the final answer.
 
 Task changes do not end wait_agent calls, add inbox messages, or start idle agents. Use send_message when another agent needs to act on a task.