Classic martini is stirred only, not shaken. shaken martini is a totaly diffrent coctail, it something that including "martini" and other alcohol, new or local inventions.

 The coctail that called martini is Jin with vermouth (usualy martini but it can be also other vermouth) served in martini glass, like coctail glass but smaller, the glass must be cold so the bartender put ice in it before starting to make the coctail, or another trik is to put the glasses in the frizer.

A stir coctail is not getting a lot of ice efect as the shaken one, there are drinks that if they will be served with ice or will be shaken with ice will be rulned very quiqly, the martini coctailes are those drinks.