### 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)

"""
        Pause/Un-pause a DAG.
        :param is_paused: Is the DAG paused
        :param including_subdags: whether to include the DAG's subdags
        :param session: session
        """
        filter_query = [
            DagModel.dag_id == self.dag_id,
        ]
        if including_subdags:
            filter_query.append(DagModel.root_dag_id == self.dag_id)
        session.execute(
            update(DagModel)
            .where(or_(*filter_query))
            .values(is_paused=is_paused)
            .execution_options(synchronize_session="fetch")
        )
        session.commit()
    @classmethod
    @internal_api_call
    @provide_session
    def deactivate_deleted_dags(