I can't believe nobody's mentioned Winston Churchill's recipe for a dry martini:
Shake gin in a cocktail shaker with cracked ice. Strain into a chilled glass and merely glance at the bottle of vermouth from across the room. Garnish w/ olive.
And I should probably preface my comments by saying that although I see my own world in black & white, I'm okay with others who don't see it the same way. If they want to call a vodka martini a martini, then I'm not going to try to stop them. And if they want to eat their BBQ with sauce, that's okay too. Not usually my preference, but life's too short to sweat the small stuff.
Making a proper cocktail is very much a mixture of art and science. The perfect martini may be the most elusive of all cocktails. So many variables come into the equation. What brand of gin and vermouth do you use? Also the quality and brand of the garnish can make a big difference. Then when you finally decide on the ingredients the temperature of the gin and vermouth during the preparation is important. It is not a matter of what taste you prefer, but how does one achieve that same taste every time you make a martini. This is where science can be important. You should always measure the ingredients precisely. If you measure by eye you will obviously not achieve the same cocktail every time. Consistency is the hallmark of a great bartender. The type of ice you use is important as well. You should use clear ice made from filtered water otherwise chlorinated water flavors will taint the cocktails. Is the ice crushed, cracked or cubed? How long do you shake or stir? Are you making a single batch or multiple drinks at one time? Are you using hard ice or has the ice been sitting around at room temperature for awhile? Are your olives cold or at room temperature? When you finally make the perfect martini to suit your own tastes you need to have recorded all the variables accurately in order to make close to the same cocktail each time.
If you are like me you aren't that picky because variety is the spice of life and life is too short to be so compulsive about any cocktail really. To me gin is one of man's greatest distillates. Vodka in comparison is the white bread of distillates and should not qualify as a martini ingredient.