### 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)

self,
        start_date: pendulum.DateTime,
        num: int | None = None,
        end_date: datetime | None = None,
    ) -> list[datetime]:
        message = "`DAG.date_range()` is deprecated."
        if num is not None:
            warnings.warn(message, category=RemovedInAirflow3Warning, stacklevel=2)
            with warnings.catch_warnings():
                warnings.simplefilter("ignore", RemovedInAirflow3Warning)
                return utils_date_range(
                    start_date=start_date, num=num, delta=self.normalized_schedule_interval
                )
        message += " Please use `DAG.iter_dagrun_infos_between(..., align=False)` instead."
        warnings.warn(message, category=RemovedInAirflow3Warning, stacklevel=2)
        if end_date is None:
            coerced_end_date = timezone.utcnow()
        else:
            coerced_end_date = end_date
        it = self.iter_dagrun_infos_between(start_date, pendulum.instance(coerced_end_date), align=False)
        return [info.logical_date for info in it]
    def is_fixed_time_schedule(self):