### 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
StartContext startContext,
        ActionListener<StartContext> listener
    ) {
        String destIndex = startContext.config.getDest().getIndex();
        SearchRequest destEmptySearch = new SearchRequest(destIndex);
        destEmptySearch.source().size(0);
        destEmptySearch.allowPartialSearchResults(false);
        ClientHelper.executeWithHeadersAsync(
            startContext.config.getHeaders(),
            ClientHelper.ML_ORIGIN,
            parentTaskClient,
            SearchAction.INSTANCE,
            destEmptySearch,
            ActionListener.wrap(searchResponse -> {
                if (searchResponse.getHits().getTotalHits().value > 0) {
                    listener.onFailure(ExceptionsHelper.badRequestException("dest index [{}] must be empty", destIndex));
                } else {
                    listener.onResponse(startContext);
                }
            }, e -> {
                if (ExceptionsHelper.unwrapCause(e) instanceof IndexNotFoundException) {
                    listener.onResponse(startContext);
                } else {
                    listener.onFailure(e);
                }
            })
        );
    }
    private void waitForAnalyticsStarted(
        PersistentTasksCustomMetadata.PersistentTask<TaskParams> task,