### Description

This applies to the following APIs:
- [x] open anomaly detection
- [x] start datafeed
- [x] start data frame analytics job
- [x] start trained model deployment

Currently, if the operation takes longer than the configured `timeout`, the response status code is `500`.

With Kibana in mind, a `4xx` response would be more suitable as it will lead to showing more useful information to the user about why the request failed.

Looking at [RFC 9110](https://www.rfc-editor.org/rfc/rfc9110.html#name-408-request-timeout), it seems the most appropriate `4xx` code would be `408 Request Timeout`, even though the RFC describes a different situation. 

The alternative, `504 Gateway Timeout`, would be more appropriate for this situation but it wouldn't improve the user experience in Kibana.
x-pack/plugin/ml/qa/native-multi-node-tests/src/javaRestTest/java/org/elasticsearch/xpack/ml/integration/MlJobIT.java
"is_hidden": true,
                  "filter": {
                    "term": {
                      "%s": "%s"
                    }
                  }
                }
              }
            }""".formatted(AnomalyDetectorsIndex.jobResultsAliasedName(jobId), Job.ID, jobId));
        client().performRequest(extraIndex2);
        
        String indicesBeforeDelete = EntityUtils.toString(
            client().performRequest(new Request("GET", "/_cat/indices/" + AnomalyDetectorsIndexFields.RESULTS_INDEX_PREFIX + "*"))
                .getEntity()
        );
        assertThat(indicesBeforeDelete, containsString(indexName));
        assertThat(indicesBeforeDelete, containsString(indexName + "-001"));
        assertThat(indicesBeforeDelete, containsString(indexName + "-002"));
        
        Request createDoc0 = new Request("PUT", indexName + "/_doc/" + 123);
        createDoc0.setJsonEntity(String.format(Locale.ROOT, """
            {"job_id":"%s", "timestamp": "%s", "bucket_span":%d, "result_type":"record"}""", jobId, 123, 1));
        client().performRequest(createDoc0);
        Request createDoc1 = new Request("PUT", indexName + "-001/_doc/" + 123);
        createDoc1.setEntity(createDoc0.getEntity());
        client().performRequest(createDoc1);
        Request createDoc2 = new Request("PUT", indexName + "-002/_doc/" + 123);
        createDoc2.setEntity(createDoc0.getEntity());
        client().performRequest(createDoc2);