Excuse me, I have this argument with another freak all the time. =oP
A proper Martini is made with about a quart of ice, 8 oz. of Stolichnaya, and your shake a shot of vermouth onto the ice, then strain it unto the ice is just ice.. then you shake the vodka in, and strain it into a Martini glass with 2 olives on a toothpick without a windmill or any flair. The toothpick is hickory, and yes it matters.
It's vodka. I grew up near pine trees. It's vodka.. gin is for tonic, and when you need a fresh smell in your house, just like that horrid Pine-Sol.
(although I like gin and tonic.. err.. good gin and tonic)
my fave Martini is prepared as you say with the martini condensing on the ice inside the shaker (dry Martini) and the bartender throws away everything which does not stick to the ice (essential oils etc.) and then vodka.. or gin.. and a twist of lemon or olive.. depending on the mood and availability..
the martini in this case is there to cover the taste of alcohol, which makes plain gin taste and smell a bit cheap..
the original martini dry had a bit more vermouth mixed-in and was originally made with french vermouth.. even though Vermouth was really invented in Italy! Hic!...