A martini is gin and vermouth, maybe with some bitters if you like
A vodka martini is vodka and vermouth, bitters again optional
A vesper martini is gin, vodka, and lillet blanc
Any of the above can be made "dirty" with olive brine if you want
Anything else is a cocktail in a martini glass. No shade if you like apple schnapps, lemon juice, and vodka, drink what you like, but it's not a martini.