The search field is defined on the main page. The default is .master, it doesn't matter which layout you choose, the elements defined on the main page will still be in one place. In your case, it turns out that the search box does not exist on your host and should be added again.
You will need to change the main page of your site and add the appropriate SP control to search in the appropriate place on your page. This will ensure that the SP search box is displayed at that location for each page.
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