### Apache Airflow Provider(s)

amazon

### Versions of Apache Airflow Providers

apache-airflow-providers-amazon==7.1.0

### Apache Airflow version

2.5.1

### Operating System

Ubuntu 18.04

### Deployment

Official Apache Airflow Helm Chart

### Deployment details

_No response_

### What happened

When looking at logs in the UI for a running task when using remote s3 logging, the logs for the task are only uploaded to s3 after the task has completed. The the `S3TaskHandler` falls back to the local logs stored on the worker in that case (by falling back to the `FileTaskHandler` behavior) and prepends the line `*** Falling back to local log` to those logs.

This is mostly fine, but for the new log streaming behavior, this means that `*** Falling back to local log` is returned from `/get_logs_with_metadata` on each call, even if there are no new logs.

### What you think should happen instead

I'd expect the falling back message only to be included in calls with no `log_pos` in the metadata or with a `log_pos` of `0`.

### How to reproduce

Start a task with `logging.remote_logging` set to `True` and `logging.remote_base_log_folder` set to `s3://something` and watch the logs while the task is running. You'll see `*** Falling back to local log` printed every few seconds.

### Anything else

_No response_

### Are you willing to submit PR?

- [X] 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)

super().__init__(base_log_folder, filename_template)
        self.remote_base = s3_log_folder
        self.log_relative_path = ""
        self._hook = None
        self.closed = False
        self.upload_on_close = True
    @cached_property
    def hook(self):
        """Returns S3Hook."""
        return S3Hook(
            aws_conn_id=conf.get("logging", "REMOTE_LOG_CONN_ID"), transfer_config_args={"use_threads": False}
        )
    def set_context(self, ti):
        super().set_context(ti)
        
        
        full_path = self.handler.baseFilename
        self.log_relative_path = pathlib.Path(full_path).relative_to(self.local_base).as_posix()
        is_trigger_log_context = getattr(ti, "is_trigger_log_context", False)
        self.upload_on_close = is_trigger_log_context or not ti.raw
        
        
        if self.upload_on_close:
            with open(self.handler.baseFilename, "w"):
                pass
    def close(self):