For a a drinks writer to write,
"I believe a drink should be at least based on gin or vodka to properly be termed a Martini, and ideally should also include vermouth"
would be like a food writer writing,
"I believe a dish should be at least based on pasta or bread to be properly termed Penne all'Arrabiata, and ideally should also include tomato."
It is evidence of a mind ignorant of its own ignorance and grasping at image.
I consider the Martini cocktail to be made from the following specific ingredients: tom gin, sweet vermouth, orange additive bitters and method ice - stirred, strained and garnished with twisted lemon zest.
I must note here that, at first, the Martini Cocktail was simply a (possibly errant) renaming of the Martinez Cocktail. I haven't come across any source listing both as separate drinks until 1930. Not all the old books agree on the garniture or the type of additive bitters. I have chosen a compromise between William Boothby's 1891 recipe and George Kappeler's 1895 recipe. Those are the oldest two recipes under the 'Martini' name I could find that omit both the sugar syrup and the liqueur found in even older recipes. Today, those oldest versions are called the Martinez Cocktail.
A different cocktail made of dry gin, dry vermouth, method ice and orange additive bitters is either a Marguerite cocktail (from 1904 with twisted lemon zest) or a Good Times cocktail (from about 1914 with a pickled green olive). I would call the vodka versions the Vodka Times cocktail and the Vodkuerite Cocktail.
As early as 1914, there was the Cat - dry gin and dry vermouth stirred through ice (without bitters), strained and garnished with a pickled green olive. The vodka version should be called a Vodka Cat - but the perennial parade of middle-class social insecurity and general ignorance has led to its admittance to the club of things mis-called "Martini." You can be a Martini, too! And then people will think you are sophisticated (and not just a drunk with a boring job).
Most conceivable combinations of spirits, aromatized wines, bitters and garniture were made and given their own names long ago. There is no excuse for a drinks writer not knowing them. What is worse is lumping them all together under one trendy name - and then pretending to be a purist or a traditionalist about it!