### 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)

raise ValueError(
                    f"A {run_type.value} DAG run cannot use ID {run_id!r} since it "
                    f"is reserved for {inferred_run_type.value} runs"
                )
        elif run_type and logical_date is not None:  
            run_id = self.timetable.generate_run_id(
                run_type=run_type, logical_date=logical_date, data_interval=data_interval
            )
        else:
            raise AirflowException(
                "Creating DagRun needs either `run_id` or both `run_type` and `execution_date`"
            )
        regex = airflow_conf.get("scheduler", "allowed_run_id_pattern")
        if run_id and not re2.match(RUN_ID_REGEX, run_id):
            if not regex.strip() or not re2.match(regex.strip(), run_id):
                raise AirflowException(
                    f"The provided run ID '{run_id}' is invalid. It does not match either "
                    f"the configured pattern: '{regex}' or the built-in pattern: '{RUN_ID_REGEX}'"
                )
        
        copied_params = copy.deepcopy(self.params)
        copied_params.update(conf or {})
        copied_params.validate()
        run = DagRun(
            dag_id=self.dag_id,
            run_id=run_id,
            execution_date=logical_date,
            start_date=start_date,
            external_trigger=external_trigger,