Those are third-party numbers. The "trillions" statement came from Google, where nobody conflates searches on Gmail with those on Web search. Your original message implied that Google inflates search numbers; I'll only say that that's a major assumption to make. (I used to know the exact numbers.)
Powered By Reporting is listed as any search that is handled by Google or Bing. "Google’s “powered by” share is composed of searches conducted at "Google entities", as well as searches on AOL and Ask’s MyWebSearch "[3]
In Comscores qSearch product that you can subscribe to, when you hit "Google Sites" it provides a drop down detailing the breakdown of all the properties. Search is hard to define when companies get to that size, and searches on other Google properties are indeed 'searches' but they are also different than searches on DuckDuckGo.
[2]http://www.comscore.com/Insights/Rankings/comScore-Releases-... [3]http://www.comscore.com/Insights/Blog/comScore-September-201...