### 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)

"""
        Return a list of dates between the interval received as parameter using this dag's schedule interval.
        Returned dates can be used for execution dates.
        :param start_date: The start date of the interval.
        :param end_date: The end date of the interval. Defaults to ``timezone.utcnow()``.
        :return: A list of dates within the interval following the dag's schedule.
        """
        warnings.warn(
            "`DAG.get_run_dates()` is deprecated. Please use `DAG.iter_dagrun_infos_between()` instead.",
            category=RemovedInAirflow3Warning,
            stacklevel=2,
        )
        earliest = timezone.coerce_datetime(start_date)
        if end_date is None:
            latest = pendulum.now(timezone.utc)
        else:
            latest = timezone.coerce_datetime(end_date)
        return [info.logical_date for info in self.iter_dagrun_infos_between(earliest, latest)]
    def normalize_schedule(self, dttm):