It is apparant that alert() and prompt() are defined in the browser simply from viewing the different way (appearance) in which they are rendered in different browsers.
What practical difference that makes eludes me. But my point was that alerts are obsolete and should not be used. That is true however they originate, DOM method or not.
Edit: alert() may well be defined in the browser but like Airblader I do not think that makes it part of the HTML DOM. So I think that the statement that alert() is a method of the DOM is not correct. I agree with Airblader - It may be part of every DOM's implementation, but it is not part of the DOM.