### 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)

os.environ["CONFIG_PROCESSOR_MANAGER_LOGGER"] = "True"
        os.environ["AIRFLOW__LOGGING__COLORED_CONSOLE_LOG"] = "False"
        
        
        
        
        
        
        
        
        
        
        importlib.reload(import_module(airflow.settings.LOGGING_CLASS_PATH.rsplit(".", 1)[0]))  
        importlib.reload(airflow.settings)
        airflow.settings.initialize()
        del os.environ["CONFIG_PROCESSOR_MANAGER_LOGGER"]
        processor_manager = DagFileProcessorManager(
            dag_directory=dag_directory,
            max_runs=max_runs,
            processor_timeout=processor_timeout,
            dag_ids=dag_ids,
            pickle_dags=pickle_dags,
            signal_conn=signal_conn,
            async_mode=async_mode,
        )
        processor_manager.start()
    def heartbeat(self) -> None: