I'm looking for suggestions on how an iPad app that includes web browser controls can distinguish these controls from ones that may be provided by the web content.
Here the chrome toolbar on top is compared with various controls that may be provided by websites. Notice how there are two search boxes in each screenshot. Sometimes there would be two "hamburger" buttons for extra navigation options.
If I'm making my own browser app, what can I do to make sure the users understand the difference?
I think you'd be surprised how smart users are, most of them are able to quickly distinguish native controls from web controls.
That being said, as long as you don't deliberately try to make your controls look like part of the browser you should be ok (this is an anti-pattern that malware often uses to imitate system controls). In the Facebook and BBC examples you provide, you'll also notice that they use their logo in the masthead, so any user quickly scanning the page will quickly associate the controls with the context due to the proximity and closure.
I'd recommend Strengthening behavioral cues in UX web design with Gestalt principles for a quick primer.
Apart from what has been said. If your web site is more of an application rather than a typical information site, you could additionally try to encourage your users to make a shortcut on the springboard. You can even provide a custom icon then and the app will run full screen. Although only in Safari.
Besides that Benjamin is right. Don't underestimate your users
