### 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)

"""Returns dataset dependencies graph."""
        nodes_dict: dict[str, Any] = {}
        edge_tuples: set[dict[str, str]] = set()
        for dag, dependencies in SerializedDagModel.get_dag_dependencies().items():
            dag_node_id = f"dag:{dag}"
            if dag_node_id not in nodes_dict and len(dependencies) > 0:
                for dep in dependencies:
                    if dep.dependency_type == "dag" or dep.dependency_type == "dataset":
                        nodes_dict[dag_node_id] = node_dict(dag_node_id, dag, "dag")
                        if dep.node_id not in nodes_dict:
                            nodes_dict[dep.node_id] = node_dict(
                                dep.node_id, dep.dependency_id, dep.dependency_type
                            )
                        if dep.source != "dataset":
                            edge_tuples.add((f"dag:{dep.source}", dep.node_id))
                        if dep.target != "dataset":
                            edge_tuples.add((dep.node_id, f"dag:{dep.target}"))
        nodes = list(nodes_dict.values())
        edges = [{"source": source, "target": target} for source, target in edge_tuples]
        data = {
            "nodes": nodes,
            "edges": edges,
        }
        return (
            htmlsafe_json_dumps(data, separators=(",", ":"), dumps=flask.json.dumps),
            {"Content-Type": "application/json; charset=utf-8"},
        )
    @expose("/object/datasets_summary")
    @auth.has_access([(permissions.ACTION_CAN_READ, permissions.RESOURCE_DATASET)])
    def datasets_summary(self):