### 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)

with create_session() as session:
            session.query(DagRun).filter(DagRun.dag_id == dag_id).delete(synchronize_session=False)
            session.query(TI).filter(TI.dag_id == dag_id).delete(synchronize_session=False)
            session.query(TaskFail).filter(TaskFail.dag_id == dag_id).delete(synchronize_session=False)
    @staticmethod
    def _occur_before(a, b, list_):
        """
        Assert that a occurs before b in the list.
        """
        a_index = -1
        b_index = -1
        for i, e in enumerate(list_):
            if e.task_id == a:
                a_index = i
            if e.task_id == b:
                b_index = i
        return 0 <= a_index < b_index
    def test_params_not_passed_is_empty_dict(self):
        """
        Test that when 'params' is _not_ passed to a new Dag, that the params
        attribute is set to an empty dictionary.
        """
        dag = models.DAG("test-dag")
        assert isinstance(dag.params, ParamsDict)
        assert 0 == len(dag.params)
    def test_params_passed_and_params_in_default_args_no_override(self):