### 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)

DagContext.pop_context_managed_dag()
    
    @staticmethod
    def _upgrade_outdated_dag_access_control(access_control=None):
        """
        Look for outdated dag level actions in DAG access_controls and replace them with updated actions.
        For example, in DAG access_control {'role1': {'can_dag_read'}} 'can_dag_read'
        will be replaced with 'can_read', in {'role2': {'can_dag_read', 'can_dag_edit'}}
        'can_dag_edit' will be replaced with 'can_edit', etc.
        """
        if not access_control:
            return None
        new_perm_mapping = {
            permissions.DEPRECATED_ACTION_CAN_DAG_READ: permissions.ACTION_CAN_READ,
            permissions.DEPRECATED_ACTION_CAN_DAG_EDIT: permissions.ACTION_CAN_EDIT,
        }
        updated_access_control = {}
        for role, perms in access_control.items():
            updated_access_control[role] = {new_perm_mapping.get(perm, perm) for perm in perms}
        if access_control != updated_access_control:
            warnings.warn(
                "The 'can_dag_read' and 'can_dag_edit' permissions are deprecated. "
                "Please use 'can_read' and 'can_edit', respectively.",
                RemovedInAirflow3Warning,
                stacklevel=3,
            )
        return updated_access_control
    def date_range(