### Apache Airflow version

Other Airflow 2 version (please specify below)

### What happened

Airflow 2.3.3

Raising the `AirflowSkipException` in the source code, using the `DockerOperator`, is supposed to mark the task as skipped, according to the [docs](https://airflow.apache.org/docs/apache-airflow/stable/concepts/tasks.html#special-exceptions). However, what happens is that the task is marked as failed with the logs showing `ERROR - Task failed with exception`.

### What you think should happen instead

Tasks should be marked as skipped, not failed.

### How to reproduce

Raise the `AirflowSkipException` in the python source code, while using the `DockerOperator`.

### Operating System

Ubuntu 20.04.5 LTS (GNU/Linux 5.4.0-125-generic x86_64)

### Versions of Apache Airflow Providers

_No response_

### Deployment

Docker-Compose

### 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)

"You can set `mount_tmp_dir` parameter to False to disable mounting and remove the warning"
            )
            assert warning_message in caplog.messages
        self.client_mock.create_container.assert_has_calls(
            [
                call(
                    command="env",
                    name="test_container",
                    environment={
                        "AIRFLOW_TMP_DIR": TEST_AIRFLOW_TEMP_DIRECTORY,
                        "UNIT": "TEST",
                        "PRIVATE": "MESSAGE",
                        "ENV": "FILE",
                        "VAR": "VALUE",
                    },
                    host_config=self.client_mock.create_host_config.return_value,
                    image=TEST_IMAGE,
                    user=None,
                    entrypoint=["sh", "-c"],
                    working_dir="/container/path",
                    tty=True,
                    hostname=None,
                ),
                call(
                    command="env",
                    name="test_container",
                    environment={"UNIT": "TEST", "PRIVATE": "MESSAGE", "ENV": "FILE", "VAR": "VALUE"},
                    host_config=self.client_mock.create_host_config.return_value,
                    image=TEST_IMAGE,
                    user=None,