### 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)

"""Scan at fix internal DAGs which are no longer present in files."""
        now = timezone.utcnow()
        elapsed_time_since_refresh = (now - self.last_deactivate_stale_dags_time).total_seconds()
        if elapsed_time_since_refresh > self.parsing_cleanup_interval:
            last_parsed = {
                fp: self.get_last_finish_time(fp) for fp in self.file_paths if self.get_last_finish_time(fp)
            }
            DagFileProcessorManager.deactivate_stale_dags(
                last_parsed=last_parsed,
                dag_directory=self.get_dag_directory(),
                stale_dag_threshold=self.stale_dag_threshold,
            )
            self.last_deactivate_stale_dags_time = timezone.utcnow()
    @classmethod
    @internal_api_call
    @provide_session
    def deactivate_stale_dags(