### 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)

def next_dagrun_info(self, last_automated_data_interval, restriction):
            if last_automated_data_interval is None:
                return DagRunInfo.interval(start, end)
            raise RuntimeError("this fails")
    dag = DAG(
        dag_id="test_iter_dagrun_infos_between_error",
        start_date=DEFAULT_DATE,
        timetable=FailingAfterOneTimetable(),
    )
    iterator = dag.iter_dagrun_infos_between(earliest=start, latest=end, align=True)
    with caplog.at_level(logging.ERROR):
        infos = list(iterator)
    
    assert infos == [DagRunInfo.interval(start, end)]
    assert caplog.record_tuples == [
        (
            "airflow.models.dag.DAG",
            logging.ERROR,
            f"Failed to fetch run info after data interval {DataInterval(start, end)} for DAG {dag.dag_id!r}",
        ),
    ]
    assert caplog.records[0].exc_info is not None, "should contain exception context"
@pytest.mark.parametrize(
    "logical_date, data_interval_start, data_interval_end, expected_data_interval",
    [
        pytest.param(None, None, None, None, id="no-next-run"),
        pytest.param(
            DEFAULT_DATE,
            DEFAULT_DATE,
            DEFAULT_DATE + timedelta(days=2),