### 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)

"""A list of columns to show in the Add form.
        This dynamically calculates additional fields from providers and add
        them to the backing list. This calculation is done exactly once (by
        checking we're referencing the class-level variable instead of the
        instance-level), and only after we enter the request context (to skip
        superfuluous checks done by Flask-Appbuilder on startup).
        """
        if self._add_columns is type(self)._add_columns and has_request_context():
            self._add_columns = [*self._add_columns, *(k for k, _ in self._iter_extra_field_names())]
        return self._add_columns
    @property
    def edit_columns(self) -> list[str]:
        """A list of columns to show in the Edit form.
        This dynamically calculates additional fields from providers and add
        them to the backing list. This calculation is done exactly once (by
        checking we're referencing the class-level variable instead of the
        instance-level), and only after we enter the request context (to skip
        superfuluous checks done by Flask-Appbuilder on startup).
        """
        if self._edit_columns is type(self)._edit_columns and has_request_context():
            self._edit_columns = [*self._edit_columns, *(k for k, _ in self._iter_extra_field_names())]
        return self._edit_columns
    @action("muldelete", "Delete", "Are you sure you want to delete selected records?", single=False)
    @auth.has_access(
        [
            (permissions.ACTION_CAN_EDIT, permissions.RESOURCE_DAG),
        ]
    )
    def action_muldelete(self, items):