If it's not in the DOM spec, it is not a DOM method. And your link is the HTML5 spec, which is not DOM. The MDN talks about Gecko DOM, which is an implementation. If Mozilla sees it as part of their DOM, that's fine. That makes it part of their DOM's implementation, not part of the DOM. If I write my own spec and call something "dom-foobar", it still isn't part of the DOM either.
So again: Unless the DOM spec itself says that alert is part of the DOM, it simply isn't, not per spec.