I'm willing to accept vodka as a variant, but the original was gin and as far as I am concerned, the only 'true' martini. And in the original recipe, the proportions included much more vermouth than is used today. It was a much smaller drink (usually about 3oz vs today's vats of booze) and the proportions were 1 part vermouth to 3 parts gin. These days, someone asks for an actual martini (instead of those super sweet chocolate or fruit cocktails that pass for 'martinis' these days) and they want you to essentially just show the bottle of vermouth to the glass, then put it away and pour straight gin (or vodka) into a glass made for a giant.
When I order a martini, I do specify a "vodka martini" with the expectation that if I ask for just "a martini," I'll get gin. With the exception of a shot of gin with a little grenadine around the holidays (tastes like Christmas) I'm not much a fan of drinking my pine trees. Unfortunately, it's hard to find anyone up here that even knows
how to make a martini, these days. I've had to explain it more than once, with mixed results.
When I make one at home, it's 5+1 Vodka and extra-dry vermouth, half a splash of brine (half-dirty, if you will) in the glass and swirled, usually with garlic-stuffed olives because they're amazing. Agitated vigorously with ice in a shaker or suitable daylight tank for 3o seconds flat, then promptly strained into the glass with three olives on a spear. The extra time does dilute it a bit, which is good for me because I'll wind up drinking it more slowly overall. I also like the ice flecks.
(That's probably an Alaska thing: we consume the most ice cream in the country per capita, and most of it in December through February. We're weird about frozen things.)