If you choose to have a search box it is better to guide the site visitor rather then restrict them, enter under Settings/Search..under Labels there is a section to add the label for radius, address, address placeholder etc. In my placeholder I have “Enter address or zip code.” It would not restrict them to those alternatives if they wanted to type in a City for instance.
You can also have a Search form title.
You can further fine tune the search form with CSS, the search layout is also determined by the plugin style you have selected.
Without restricting , normally a search is by address but a city, state or other name typed in a search box will work unless you are using a Do not allow selector and address for example.
The technical detail of how search works, at least how the basic search always worked before there have been additional features added for premier such as what you are using for category selectors..info and tech details how search works , can be found here