There's a difference between the search module and the search skin object.
What you are refrring to is the skin obejct, the search box you see inside the skin.
When you use the search input module, that's located at:
\DesktopModules\Admin\SearchInput

To complicate things you should not edit these files, but create a copy of the module or skin object and  edit that.
The reason why you should not do these core modifications is that you would have to place them back on every upgrade.
It's better to work on a copy.

BTW, the Search skin object has a lot of options to show / hide certain parts of what it renders.
HTH