In addition to what magicalist said, here is some more information:
The Firefox Search widget is indeed installed by default but not activated by default. You can use it either by adding it to the home screen or via the “swipeup” gesture.
A problem is that it seems to have its own “Clear search history” and does not obey the one from the Firefox privacy settings.
Another problem is that you may activate it and set it as permanent default for the “swipeup” gesture by accident, and then there seems to be no way to change it to something else unless you install some other application that provides a “swipeup” action, in which case you are presented again with the interface to choose your default swipeup action.
This is a general problem in Android. An app that can help you change easily the default swipeup action is the Swipeup Utility:
You can use Swipeup Utility to set the default swipeup action to Nothing, or you can use it to clear the default of the swipeup gesture, so that you are a presented with the interface to chose a default again.
Last, the Firefox Search widget does not allow only Google Search. It is set to whatever your default search provider is in Firefox.
Mine defaults to Yahoo, which is exactly what you'd expect from the Firefox Search app. It's just a launcher intent that was registered when you installed Firefox.
According to the bottom box on that page, your search history should have been cleared from the steps you took. Sounds like a bug.
edit: you can prove this to yourself by going to Settings -> Apps -> Firefox, scrolling down to the "Launch By Default" section and clicking the "Clear Defaults" button. The next time you drag up from the bottom, your phone will ask you what you want to launch by default (assuming you still have other launchers, e.g. the Google App installed).
(This is on Android, BTW.)