### 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/src/main/java/org/elasticsearch/xpack/ml/action/TransportStartDataFrameAnalyticsAction.java
private final DataFrameAnalyticsConfig config;
        private final DataFrameAnalyticsTask.StartingState startingState;
        private volatile ExtractedFields extractedFields;
        private StartContext(DataFrameAnalyticsConfig config, List<PhaseProgress> progressOnStart) {
            this.config = config;
            this.startingState = DataFrameAnalyticsTask.determineStartingState(config.getId(), progressOnStart);
        }
    }
    /**
     * Important: the methods of this class must NOT throw exceptions.  If they did then the callers
     * of endpoints waiting for a condition tested by this predicate would never get a response.
     */
    private static class AnalyticsPredicate implements Predicate<PersistentTasksCustomMetadata.PersistentTask<?>> {