I believe if you use the microdata option, you are supposed to apply the itemprop, itemscope, etc attributes to your existing search form markup. The reason you see the search button is because you are essentially adding another form to the page. With that in mind you might want to use the json-ld data.
NB: I should point out that I've seen both examples of Magento sites without any meta data appearing with sitelink search and sites with meta data that don't display it in the search results. Ultimately there is a google algorithm that determines when to show it, the