### 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
IndexRequest indexRequest = new IndexRequest(sourceIndex);
            
            String docId = i == 0 ? "outlier" : "normal" + i;
            indexRequest.id(docId);
            indexRequest.source("numeric_1", i == 0 ? 100.0 : 1.0, "numeric_2", 1.0, "categorical_1", "foo_" + i);
            bulkRequestBuilder.add(indexRequest);
        }
        BulkResponse bulkResponse = bulkRequestBuilder.get();
        if (bulkResponse.hasFailures()) {
            fail("Failed to index data: " + bulkResponse.buildFailureMessage());
        }
        String id = "test_outlier_detection_with_custom_params";
        DataFrameAnalyticsConfig config = buildAnalytics(
            id,
            sourceIndex,
            sourceIndex + "-results",
            null,
            new OutlierDetection.Builder().setNNeighbors(3)
                .setMethod(OutlierDetection.Method.DISTANCE_KNN)
                .setFeatureInfluenceThreshold(0.01)
                .setComputeFeatureInfluence(false)
                .setOutlierFraction(0.04)
                .setStandardizationEnabled(true)
                .build()
        );
        putAnalytics(config);
        assertIsStopped(id);
        assertProgressIsZero(id);
        startAnalytics(id);
        waitUntilAnalyticsIsStopped(id);