This is always a great debate... I especially love how when you ask people what mirepoix is, they name three vegetables, instead of answering the actual question as to what mirepoix is. Its not quite as defined as naming three vegetables, but the answer is really actually a bit simpler than that. Mirepoix is simply defined as a rough garnish of aromatic vegetables used to flavor a stock, sauce, etc(rice, stews, soups, so on and so forth). It can be strained out, or used in the final presentation of the end product.
The vegetables used is determined by the context of the preparation. For a brown veal stock, the standard onion/celery/carrot is what you would likely use. For a white veal stock, carrots would not be used, and you would likely instead opt for leeks and/or some parsnips. For a seafood fumet, carrots,again, would likely be left out, and in addition to the aforementioned leeks and parsnips, fennel is often used. As for no cooking of the mirepoix prior to going into the stock, thats not entirely accurate. For a brown stock, you're very likely roasting(or sauteeing) those vegetables to caramelize them and create a bolder, more pronounced flavor. With your white stocks, they can simply be added right into the stock, or are lightly sweated.
Finally, for presentation, in stocks you're obviously straining these vegetables out prior to using it. But for a sort of "country" style beef stew where you're going for a less refined presentation, its obviously very acceptable to serve with the mirepoix in the stew. Obviously, given the context, the cut size/shape will very according to what you're doing. Typically in stocks, the shape doesn't really matter since its being left out- the most important thing is size of the cut(longer cooking stocks like a brown veal stock will be cut larger than a chicken stock, and fish fumets will require smaller mirepoix yet since they are generally cooked for only an hour or so). Most of us aren't that particular about how pretty mirepoix for stocks look, and then theres people like Thomas Keller, that require absolute precision and near competition worthy knife work for vegetables used for stocks, and won't hesitate to call you a piece of worthless garbage if you're knife work isn't absolutely pristine.
All in all, when it comes to mirepoix, its all about context.