### 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)

"start_date",
        "end_date",
        "duration",
        "note",
        "job_id",
        "hostname",
        "unixname",
        "priority_weight",
        "queue",
        "queued_dttm",
        "try_number",
        "pool",
        "queued_by_job_id",
        "external_executor_id",
        "log_url",
    ]
    order_columns = [
        item
        for item in list_columns
        if item
        not in [
            "try_number",
            "log_url",
            "external_executor_id",
            "note",  
        ]
    ]
    label_columns = {
        "dag_run.execution_date": "Logical Date",
    }