### 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/MlJobIT.java
assertNull(ioe.get().getMessage(), ioe.get());
        }
        assertEquals(numThreads, responses.size() + responseExceptions.size());
        
        for (ResponseException re : responseExceptions.values()) {
            assertEquals(re.getMessage(), 404, re.getResponse().getStatusLine().getStatusCode());
        }
        for (Response response : responses.values()) {
            assertEquals(EntityUtils.toString(response.getEntity()), 200, response.getStatusLine().getStatusCode());
        }
        assertNotNull(recreationResponse.get());
        assertEquals(
            EntityUtils.toString(recreationResponse.get().getEntity()),
            200,
            recreationResponse.get().getStatusLine().getStatusCode()
        );
        if (recreationException.get() != null) {
            assertNull(recreationException.get().getMessage(), recreationException.get());
        }
        String expectedReadAliasString = """
            "%s":{"filter":{"term":{"job_id":{"value":"%s"}}},"is_hidden":true}""".formatted(
            AnomalyDetectorsIndex.jobResultsAliasedName(jobId),
            jobId
        );
        String expectedWriteAliasString = """
            "%s":{"is_hidden":true}""".formatted(AnomalyDetectorsIndex.resultsWriteAlias(jobId));
        try {