### Apache Airflow version

2.6.3

### What happened

When running multiple standalone DAG processors with separate DAG directories using the `--subdir` argument the processors keep deactivating each other's DAGs (and reactivating their own).

After stepping through the code with a debugger I think the issue is that the calls [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L794) and [here](https://github.com/apache/airflow/blob/2.6.3/airflow/dag_processing/manager.py#L798) have no awareness of the DAG directories.

### What you think should happen instead

The DAG processors should not touch each other's DAGs in the metadata DB.

### How to reproduce

Start two or more standalone DAG processors with separate DAG directories and observe (e.g. via the UI) how the list of active DAGs keeps changing constantly.

### Operating System

Linux 94b223524983 6.1.32-0-virt #1-Alpine SMP PREEMPT_DYNAMIC Mon, 05 Jun 2023 09:39:09 +0000 x86_64 x86_64 x86_64 GNU/Linux

### Versions of Apache Airflow Providers

_No response_

### Deployment

Other Docker-based deployment

### 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)

"""Exposes a CLI specific to this DAG."""
        check_cycle(self)
        from airflow.cli import cli_parser
        parser = cli_parser.get_parser(dag_parser=True)
        args = parser.parse_args()
        args.func(args, self)
    @provide_session
    def test(
        self,
        execution_date: datetime | None = None,
        run_conf: dict[str, Any] | None = None,
        conn_file_path: str | None = None,
        variable_file_path: str | None = None,
        session: Session = NEW_SESSION,
    ) -> DagRun:
        """
        Execute one single DagRun for a given DAG and execution date.
        :param execution_date: execution date for the DAG run
        :param run_conf: configuration to pass to newly created dagrun
        :param conn_file_path: file path to a connection file in either yaml or json
        :param variable_file_path: file path to a variable file in either yaml or json
        :param session: database connection (optional)
        """
        def add_logger_if_needed(ti: TaskInstance):