### Apache Airflow version

main (development)

### What happened

https://github.com/apache/airflow/pull/27755 added the triggerer to the rest api health endpoint, but not the main one served on `/health`.

### What you think should happen instead

As documented [here](https://airflow.apache.org/docs/apache-airflow/2.6.1/administration-and-deployment/logging-monitoring/check-health.html#webserver-health-check-endpoint), the `/health` endpoint should include triggerer info like shown on `/api/v1/health`.

### How to reproduce

Compare `/api/v1/health` and `/health` responses.

### Operating System

mac os

### Versions of Apache Airflow Providers

_No response_

### Deployment

Virtualenv installation

### Deployment details

_No response_

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

dag = get_airflow_app().dag_bag.get_dag(dag_id)
            if dag:
                ti.task = dag.get_task(ti.task_id)
            if response_format == "json":
                logs, metadata = task_log_reader.read_log_chunks(ti, try_number, metadata)
                message = logs[0] if try_number is not None else logs
                return {"message": message, "metadata": metadata}
            metadata["download_logs"] = True
            attachment_filename = task_log_reader.render_log_filename(ti, try_number, session=session)
            log_stream = task_log_reader.read_log_stream(ti, try_number, metadata)
            return Response(
                response=log_stream,
                mimetype="text/plain",
                headers={"Content-Disposition": f"attachment; filename={attachment_filename}"},
            )
        except AttributeError as e:
            error_messages = [f"Task log handler does not support read logs.\n{str(e)}\n"]
            metadata["end_of_log"] = True
            return {"message": error_messages, "error": True, "metadata": metadata}
    @expose("/log")
    @auth.has_access(
        [
            (permissions.ACTION_CAN_READ, permissions.RESOURCE_DAG),
            (permissions.ACTION_CAN_READ, permissions.RESOURCE_TASK_INSTANCE),
            (permissions.ACTION_CAN_READ, permissions.RESOURCE_TASK_LOG),
        ]
    )
    @action_logging
    @provide_session
    def log(self, session: Session = NEW_SESSION):