### 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)

"""Occasionally print out stats about how fast the files are getting processed."""
        if 0 < self.print_stats_interval < time.monotonic() - self.last_stat_print_time:
            if self._file_paths:
                self._log_file_processing_stats(self._file_paths)
            self.last_stat_print_time = time.monotonic()
    @staticmethod
    @internal_api_call
    @provide_session
    def clear_nonexistent_import_errors(file_paths: list[str] | None, session=NEW_SESSION):
        """
        Clear import errors for files that no longer exist.
        :param file_paths: list of paths to DAG definition files
        :param session: session for ORM operations
        """
        query = session.query(errors.ImportError)
        if file_paths:
            query = query.filter(~errors.ImportError.filename.in_(file_paths))