First, 'dry' indicates that very little vermouth has been added to the cocktail, so the gin is the primary focus. The typical ratio is 6 parts gin to 1 part vermouth. However, order 'extra dry' and you'll get the slightest splash of vermouth, or even just a glass-coating wash.
When talking about alcoholic drinks, "dry" is just the opposite of sweet. Vermouth has a slight sweetness to it, so use less vermouth to make the drink "dry".
Personally I like my martinis "dirty" (extra olive juice), with a good amount of vermouth. Sweet and salty with the floral notes of the gin in the background. If I wanted to just taste the gin then I'd put it in a shot glass and drink it straight instead.
My city has a "no straight spirits" rule for bars, it's a failed attempt to stop binge drinking
So I would order whisky on the rocks, hold the ice
Were I a gin fan it would be a "perfectly dry martini"
The bar staff agree the rule is stupid and are usually happy to work around them. Binge drinking in bars hardly happens anyway as it is too expensive, we have very high alcohol taxes, except on wine.