### 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
assertThatAuditMessagesMatch(
            id,
            "Created analytics with type [outlier_detection]",
            "Estimated memory usage [",
            "Starting analytics on node",
            "Started analytics",
            "Creating destination index [test-outlier-detection-with-enough-docs-to-scroll-results]",
            "Started reindexing to destination index [test-outlier-detection-with-enough-docs-to-scroll-results]",
            "Finished reindexing to destination index [test-outlier-detection-with-enough-docs-to-scroll-results]",
            "Started loading data",
            "Started analyzing",
            "Started writing results",
            "Finished analysis"
        );
    }
    public void testOutlierDetectionWithMoreFieldsThanDocValueFieldLimit() throws Exception {
        String sourceIndex = "test-outlier-detection-with-more-fields-than-docvalue-limit";
        client().admin().indices().prepareCreate(sourceIndex).get();
        GetSettingsRequest getSettingsRequest = new GetSettingsRequest();
        getSettingsRequest.indices(sourceIndex);
        getSettingsRequest.names(IndexSettings.MAX_DOCVALUE_FIELDS_SEARCH_SETTING.getKey());
        getSettingsRequest.includeDefaults(true);
        GetSettingsResponse docValueLimitSetting = client().admin().indices().getSettings(getSettingsRequest).actionGet();
        int docValueLimit = IndexSettings.MAX_DOCVALUE_FIELDS_SEARCH_SETTING.get(
            docValueLimitSetting.getIndexToSettings().values().iterator().next()
        );
        BulkRequestBuilder bulkRequestBuilder = client().prepareBulk();
        bulkRequestBuilder.setRefreshPolicy(WriteRequest.RefreshPolicy.IMMEDIATE);
        for (int i = 0; i < 100; i++) {
            StringBuilder source = new StringBuilder("{");