### 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)

job_id = 42
        dag = DAG(dag_id="test_create_dagrun_job_id_is_set")
        dr = dag.create_dagrun(
            run_id="test_create_dagrun_job_id_is_set", state=State.NONE, creating_job_id=job_id
        )
        assert dr.creating_job_id == job_id
    def test_dag_add_task_checks_trigger_rule(self):
        
        from airflow.exceptions import FailStopDagInvalidTriggerRule
        from airflow.utils.trigger_rule import TriggerRule
        task_with_non_default_trigger_rule = EmptyOperator(
            task_id="task_with_non_default_trigger_rule", trigger_rule=TriggerRule.DUMMY
        )
        non_fail_stop_dag = DAG(
            dag_id="test_dag_add_task_checks_trigger_rule", start_date=DEFAULT_DATE, fail_stop=False
        )
        try:
            non_fail_stop_dag.add_task(task_with_non_default_trigger_rule)
        except FailStopDagInvalidTriggerRule as exception:
            assert (
                False
            ), f"dag add_task() raises FailStopDagInvalidTriggerRule for non fail stop dag: {exception}"
        
        from airflow.models.abstractoperator import DEFAULT_TRIGGER_RULE
        fail_stop_dag = DAG(
            dag_id="test_dag_add_task_checks_trigger_rule", start_date=DEFAULT_DATE, fail_stop=True
        )
        task_with_default_trigger_rule = EmptyOperator(
            task_id="task_with_default_trigger_rule", trigger_rule=DEFAULT_TRIGGER_RULE
        )