Butter braising seems to be accepted to have the obvious meaning, such as butter braised cabbage that's simply cooked gently in butter and the same volume of water. Really, it just shows that it's not worth getting too pedantic about what names of methods mean. I suspect that if you told most reasonably experienced cooks to "braise cabbage in butter," you'd get something pretty similar from all of them. And I doubt many, given their experience, would try to sear it and then shove it with liquid in a covered pan into the oven. But I think if you asked them to braise a pancake, they would just be puzzled.
I think most of the named methods are much older than the very flexible equipment we have today, like ready availability to burner, oven, broiler, and every manner of pot and pan. And how much methodological territory is covered by some terms is demonstrated by Escoffier's statement that braising was one of the most difficult methods to master. By which, I take him to mean that it's simple to say what braising is but difficult to apply it to many different foods to produce precise and subtle effects. The carrot that happens to ride along with the braised pork is hardly an attractive or particularly flavorful carrot. Now, try braising the carrots alone for a dish that would be noteworthy. Takes a lot more experience and knowledge than just knowing the brief description of braising.
And because we have all these options of equipment and cookware, we can, if we have enough experience (of luck) to do it, combine methods and parts of methods in ways that just don't fit definitions.
And why the cabbages are said to be braised, but mushrooms cooked in an open pan with butter in a hot oven are said to be broiled, I'm not at all sure. Besides. No one complains, because anything's good cooked in butter. A common art gum eraser broiled in butter is good, as escargot demonstrates.