### 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)

conf=request_conf,
                form=form,
                is_dag_run_conf_overrides_params=is_dag_run_conf_overrides_params,
            )
        flash(f"Triggered {dag_id}, it should start any moment now.")
        return redirect(origin)
    def _clear_dag_tis(
        self,
        dag: DAG,
        start_date: datetime.datetime | None,
        end_date: datetime.datetime | None,
        *,
        origin: str | None,
        task_ids: Collection[str | tuple[str, int]] | None = None,
        recursive: bool = False,
        confirmed: bool = False,
        only_failed: bool = False,
        session: Session,
    ):
        if confirmed:
            count = dag.clear(
                start_date=start_date,
                end_date=end_date,
                task_ids=task_ids,
                include_subdags=recursive,
                include_parentdag=recursive,
                only_failed=only_failed,
                session=session,
            )
            msg = f"{count} task instances have been cleared"