Include examples from integrationtests/ui in example test run

A framework to run and test whether demo examples are working has been added in #853. At the moment it includes examples from `examples/demo` and `examples/tutorials/doc_examples` folders.

There are a few more "examples" in `integrationtests/ui` folder that were written with testing in mind and it might be good to include them in the test run. Most of these examples are old, so curation might be needed before they can be added.