So guys... sit back, relax, and get ready because I'm about to nerd out and learn you some shit:
The real difference between shaking a martini (or and drink for that matter) and stirring is purely about incorporating air into the drink or not.
There is no real 'right way' but a purist will always avoid shaking a drink like a martini so as not to "bruise" the spirit, bruising is the slight color change and the difference in taste, a shaken martini will have a slightly clouded look - hence the term bruised- and will taste slightly sharper due to the air bubbles that are forced into the drink whereas a stirred drink well be clear and hold a much "truer" taste of the spirit being used due to the water from the ice slowly melting and only slightly diluting the drink without lending too much taste.
The real difference between shaking a martini (or and drink for that matter) and stirring is purely about incorporating air into the drink or not.
There is no real 'right way' but a purist will always avoid shaking a drink like a martini so as not to "bruise" the spirit, bruising is the slight color change and the difference in taste, a shaken martini will have a slightly clouded look - hence the term bruised- and will taste slightly sharper due to the air bubbles that are forced into the drink whereas a stirred drink well be clear and hold a much "truer" taste of the spirit being used due to the water from the ice slowly melting and only slightly diluting the drink without lending too much taste.