### 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
NodeAcknowledgedResponse response = startAnalytics(id);
        assertThat(response.getNode(), emptyString());
        
        assertBusy(() -> {
            GetDataFrameAnalyticsStatsAction.Response.Stats stats = getAnalyticsStats(id);
            assertThat(stats.getState(), equalTo(DataFrameAnalyticsState.STARTING));
            assertThat(stats.getNode(), is(nullValue()));
            assertThat(stats.getAssignmentExplanation(), containsString("persistent task is awaiting node assignment"));
        });
        stopAnalytics(id);
        waitUntilAnalyticsIsStopped(id);
        assertThatAuditMessagesMatch(
            id,
            "Created analytics with type [outlier_detection]",
            "Estimated memory usage [",
            "Job requires at least [1tb] free memory on a machine learning node to run",
            "Started analytics",
            "Stopped analytics"
        );
    }
    public void testOutlierDetectionStopAndRestart() throws Exception {
        String sourceIndex = "test-outlier-detection-stop-and-restart";
        client().admin()
            .indices()
            .prepareCreate(sourceIndex)
            .setMapping("numeric_1", "type=double", "numeric_2", "type=float", "categorical_1", "type=keyword")
            .get();
        BulkRequestBuilder bulkRequestBuilder = client().prepareBulk();
        bulkRequestBuilder.setRefreshPolicy(WriteRequest.RefreshPolicy.IMMEDIATE);