### 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)

"Using forward slash ('/') in a DAG run ID is deprecated. Note that this character "
                "also makes the run impossible to retrieve via Airflow's REST API.",
                "warning",
            )
        run_conf = {}
        if request_conf:
            try:
                run_conf = json.loads(request_conf)
                if not isinstance(run_conf, dict):
                    flash("Invalid JSON configuration, must be a dict", "error")
                    form = DateTimeForm(data={"execution_date": execution_date})
                    return self.render_template(
                        "airflow/trigger.html",
                        form_fields=form_fields,
                        dag_id=dag_id,
                        origin=origin,
                        conf=request_conf,
                        form=form,
                        is_dag_run_conf_overrides_params=is_dag_run_conf_overrides_params,
                        recent_confs=recent_confs,
                    )
            except json.decoder.JSONDecodeError:
                flash("Invalid JSON configuration, not parseable", "error")
                form = DateTimeForm(data={"execution_date": execution_date})
                return self.render_template(
                    "airflow/trigger.html",
                    form_fields=form_fields,
                    dag_id=dag_id,
                    origin=origin,
                    conf=request_conf,