Users will not understand why there are two back buttons. That'd be like working around pastebin issues by including a different "copy selection" button.

They'll click the browser back button, get thrown back onto the search/new tab page, they'll click forward to get back to the webapp (which will have saved all of their state), and they'll understand just fine going forward.

First of all: user testing I've observed strongly suggests that no, they won't. You would. The average user doesn't have nearly your level of savvy.

Secondly: that's already a bad user experience in contrast to just unifying the behavior behind one button. Why did the user have to discover that a button doesn't work the way they expect? Why are they going to have to remember it doesn't work the right way now? And they'll have to repeat that experience for every web app they use? That's a mess.