### 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/RunDataFrameAnalyticsIT.java
GetResponse destDocGetResponse = client().prepareGet().setIndex(config.getDest().getIndex()).setId(hit.getId()).get();
            assertThat(destDocGetResponse.isExists(), is(true));
            Map<String, Object> sourceDoc = hit.getSourceAsMap();
            Map<String, Object> destDoc = destDocGetResponse.getSource();
            for (String field : sourceDoc.keySet()) {
                assertThat(destDoc.containsKey(field), is(true));
                assertThat(destDoc.get(field), equalTo(sourceDoc.get(field)));
            }
            assertThat(destDoc.containsKey("ml"), is(true));
            @SuppressWarnings("unchecked")
            Map<String, Object> resultsObject = (Map<String, Object>) destDoc.get("ml");
            assertThat(resultsObject.containsKey("outlier_score"), is(true));
            double outlierScore = (double) resultsObject.get("outlier_score");
            assertThat(outlierScore, allOf(greaterThanOrEqualTo(0.0), lessThanOrEqualTo(1.0)));
        }
        assertProgressComplete(id);
        assertThat(searchStoredProgress(id).getHits().getTotalHits().value, equalTo(1L));
        assertThatAuditMessagesMatch(
            id,
            "Created analytics with type [outlier_detection]",
            "Estimated memory usage [",
            "Starting analytics on node",
            "Started analytics",
            "Creating destination index [test-outlier-detection-with-more-fields-than-docvalue-limit-results]",
            "Started reindexing to destination index [test-outlier-detection-with-more-fields-than-docvalue-limit-results]",
            "Finished reindexing to destination index [test-outlier-detection-with-more-fields-than-docvalue-limit-results]",
            "Started loading data",
            "Started analyzing",
            "Started writing results",
            "Finished analysis"