Exposed filter : search term with select which 'field' to search by
I have created a custom module inspired by your question. You can find it here

It is not created/approved as a full project on Drupal.org. But it will approved soon. Download the module, enable it. Under filter add a filter that says Exposed fields Filter then select the fields you want to expose to visitors. Only those field will appear to visitors which you select here.

I hope it helps you for now.