The martini in its classic form was made with gin -- make no mistake about it -- and years ago a martini meant gin, period. Vodka has usurped gin for many years now as the base of choice, so that has to be considered in this discussion. It should also be understood that a classic martini contained dry vermouth and perhaps orange bitters, but we will leave the bitters out for now.
Now I am not here to tell anyone what proportions of vermouth to have, or whether you should order it shaken or not. I simply want to enlighten those of you who were born after WWII as to how it was done then, and how easy it is to get what you want by following these simple steps:
First: Call your base, either gin or vodka or your brand of choice, e.g., Beefeater or Stoli.
Second: Dry or Extra Dry or Wet (Dry=less vermouth; Extra Dry=no vermouth; Wet=extra vermouth)
Third: Choose "Up" or "On the Rocks"
Fourth: Your garnish, olives or a lemon twist. (With a cocktail onion the drink is called a Gibson, so you don't need to say onions just substitute Gibson for martini). They used to order them "Naked" if they didn't want a garnish, but I haven't heard that in a while.
Lastly: If the bartender looks like he/she knows what they are doing, and the establishment is one of high repute, you don't need to tell them whether to shake or stir, because they should know that you only shake if a cocktail contains ingredients other than booze like juices or cream, or if the customer requests it, or if the drink is called "dirty," which means with olive brine which then could be shaken).