Sorry for the delayed reply to this. Entries in your Trending Search Terms report that have their term set to status:public can be created if a user selects this search filter on the search form:
If they then click the “Search” button without entering any additional terms, a search entry will be created with its term set to status:public.
Another way the entry could be generated would be if users were clicking on a link that had that filter set, but no other search terms. For example, clicking this link will generate a search log entry with its term set to status:public: https://meta.discourse.org/search?expanded=true&q=status%3Apublic.
Is it possible that you have a link in that form somewhere on the internet that users could be clicking?