### 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)

form=form,
                is_dag_run_conf_overrides_params=is_dag_run_conf_overrides_params,
                recent_confs=recent_confs,
            )
        try:
            execution_date = timezone.parse(request_execution_date)
        except ParserError:
            flash("Invalid execution date", "error")
            form = DateTimeForm(data={"execution_date": timezone.utcnow().isoformat()})
            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,
            )
        dr = DagRun.find_duplicate(dag_id=dag_id, run_id=run_id, execution_date=execution_date)
        if dr:
            if dr.run_id == run_id:
                message = f"The run ID {run_id} already exists"
            else:
                message = f"The logical date {execution_date} already exists"
            flash(message, "error")
            return redirect(origin)
        
        if run_id and "/" in run_id:
            flash(