Custom Google Search active with standard WP search widget

I am using Custom Google Search, which I love! One of the things I noticed is that if you use the standard search widget that comes with WordPress, it still uses the Google Custom Search.

The benefit of it is that the Custom Google Search plugin does not have an option to have a title on top like most widgets do.

However, I would also like a search that just searches the website only.. Anyone aware of any working plugins for wordpress. Note: This particular website is NOT MU and does not use buddypress.