Does somebody know, whether the SiteSearchBox, like on the screenshot:
has a kind of own referrer? How is it possible to identify visitors from SiteSearchBox in Analytics?
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Indeed it knows where the visitor comes from. That is an event in Pinterest's search box on Google's page. They know and record all events and store it server side and on your machines/devices via cookies for history.
It is the same if someone were to use your search box if you are using Analytics, they would know that and store it as well. It is all a part of their success in regards to algorithms and the correct content to show you and for ranking purposes.
In the case of the screenshot, Google would be the referrer and Pinterest's search being the event.
Google knows every action and event that takes place on their sites and those giving them the right whether it be allowing to be crawled or adding Analytics to their web page.
So in short YES, they do know where the visitor comes from.