There's going to be a lot of different opinions on this but, having served thousands of martinis to the affluent, my take on it is this:
A true martini is 2oz gin and 5-6 drops of dry vermouth, stirred with ice and strained into an "up" (martini) glass and garnished with an olive. I understand that originally (or so I've been told and wiki confirms this) that a martini calls for 1/5 vermouth. In my book a "dry martini" only has 1 drop of vermouth and an "extra dry" has 1 drop swirled around the glass and "whipped out".
When a martini is ordered though, I found it unavoidable to ask "gin or vodka?" because well over half (probably 70%+) want vodka. Most seemed oblivious that a martini was made with anything other than vodka
As far as the shaved ice goes you are essentially getting to the core of what makes a "good" martini. The object is to chill the liquor as cold as you can without diluting it with water from the ice. Essentially, if you have an up glass and a bottle in the freezer all you have to do is add 1-6 drops of vermouth and your garnish (olive) and you have the "perfect martini". The shaved ice comes from shaking a martini, which also seems to be the "standard" now (as opposed to stirring a martini as a purist would). A sign of a "good" shaken martini is little shards of ice floating in the top of the cocktail after it has been poured through a strainer.
I was told, and find it true, that being a bartender is more about showmanship than anything else. My recipe for the "perfect martini" was as follows:
1) Add ice to an up glass and fill it with soda water.
2) Put my finger over the Vermouth pourer and carefully add 6 drops to the mixing tumbler.
3) Add a 2 count (2 oz) of alcohol to the mixing tumbler.
4) Add 5-6 cubes of ice to the tumbler and shake like hell for about 5 seconds, rest 2 seconds and shake like hell for another 5 seconds all while pouring the ice/water out of your glass and "whipping" the excess droplets onto the ground.
5) Strain the cocktail into the glass so that it comes nearly level with the rim and (this is what will really impress the customer) shake twice to get every last drop into the glass.
6) Set the martini in front of the customer and add your olive, which should push the martini just over the rim of the glass without any of it "bursting the bubble" and dribbling out. At this point the customer should have to lean over and sip the drink before picking it up
But yeah, all the appletinis, chocolatinis, etc. are just complete bastardizations of what is basically chilled vodka.
