[ML] Enable PUT trained model API integration tests (#189697)

## Summary

Closes #189637


### Checklist

- [x] [Flaky Test
Runner](https://ci-stats.kibana.dev/trigger_flaky_test_runner/1) was
used on any tests changed

x-pack/test/api_integration/apis/ml/trained_models/put_model.ts

@@ -15,8 +15,7 @@ export default ({ getService }: FtrProviderContext) => {
   const supertest = getService('supertestWithoutAuth');
   const ml = getService('ml');
 
-  // FLAKY: https://github.com/elastic/kibana/issues/189637
-  describe.skip('PUT trained_models', () => {
+  describe('PUT trained_models', function () {
     before(async () => {
       await ml.api.initSavedObjects();
       await ml.testResources.setKibanaTimeZoneToUTC();
@@ -38,9 +37,8 @@ export default ({ getService }: FtrProviderContext) => {
         .send(requestBody);
       ml.api.assertResponseStatusCode(200, putResponseStatus, putResponseBody);
 
-      expect(omit(putResponseBody, 'create_time')).to.eql({
+      expect(omit(putResponseBody, ['create_time', 'model_type'])).to.eql({
         model_id: 'dfa_regression_model_n_0',
-        model_type: 'tree_ensemble',
         created_by: 'api_user',
         version: '12.0.0',
         model_size_bytes: 304,