fix(host-agent): distinguish connected devices from busy-with-task in console

DeviceManager marks a device "busy" as soon as an Appium/WDA session is
connected, which is unrelated to whether a task is currently executing on
it. The Host Agent's local console displayed this raw status, making
connected-but-idle devices look permanently busy. Cross-reference the
device id against AgentStatusTracker's current_assignment (already
tracked via mark_assignment_started/finished) to show "connected" unless
a task is actually running on that device.

apps/device-host-agent/tests/test_web_app.py

@@ -1,9 +1,11 @@
 from __future__ import annotations
 
 import re
+from datetime import UTC, datetime
 
 from fastapi.testclient import TestClient
 
+from cloud.internal_api.models import AssignmentModel
 from device.manager import DeviceManager
 from host_agent.config import HostAgentConfig
 from host_agent.history import ConsoleHistoryStore
@@ -53,6 +55,7 @@ def _build_client(tmp_path, *, create_account: bool = True) -> tuple[TestClient,
         "local_account_store": local_account_store,
         "history_store": history_store,
         "session_manager": session_manager,
+        "status_tracker": status_tracker,
     }
     return client, context
 
@@ -119,6 +122,52 @@ def test_login_with_correct_password_sets_cookie_and_dashboard_succeeds(
     assert "control.example" in response.text
 
 
+def test_connected_device_without_running_task_shows_connected_not_busy(
+    tmp_path,
+) -> None:
+    client, context = _build_client(tmp_path)
+    _login(client)
+    context["manager"].register_device(
+        "device-a",
+        lambda: object(),  # type: ignore[arg-type,return-value]
+        status="busy",
+    )
+
+    dashboard_response = client.get("/")
+    api_response = client.get("/api/status")
+
+    assert "connected" in dashboard_response.text
+    assert "<td>busy</td>" not in dashboard_response.text
+    assert api_response.json()["devices"][0]["status"] == "connected"
+
+
+def test_device_running_current_assignment_still_shows_busy(tmp_path) -> None:
+    client, context = _build_client(tmp_path)
+    _login(client)
+    context["manager"].register_device(
+        "device-a",
+        lambda: object(),  # type: ignore[arg-type,return-value]
+        status="busy",
+    )
+    context["status_tracker"].mark_assignment_started(
+        AssignmentModel(
+            task_id="task-a",
+            attempt=1,
+            lease_id="lease-a",
+            lease_expires_at=datetime(2026, 7, 12, tzinfo=UTC),
+            host_id="host-a",
+            device_id="device-a",
+            goal="open settings",
+        )
+    )
+
+    dashboard_response = client.get("/")
+    api_response = client.get("/api/status")
+
+    assert "<td>busy</td>" in dashboard_response.text
+    assert api_response.json()["devices"][0]["status"] == "busy"
+
+
 def test_mutating_post_without_csrf_token_is_rejected_and_makes_no_change(
     tmp_path,
 ) -> None: