I think it only fair to let you, my dedicated reader, know my preferences.   I order Bombay Sapphire Martini shaken with just a splash of Vermouth served in a very well chilled Martini glass with a twist of lemon.

I will occasionally order Bombay Sapphire East, when available. If so I skip the Vermouth entirely. If a lime wedge is offered I occasionally choose that over lemon but the difference is very subtle and either is excellent if fresh

Things that make a Martini special, other than an abundance of gin, include having it served ice cold, with freshly peeled garnish. Martini pet peeves include a small serving, excessive Vermouth, having it mildly chilled and/or diluted, and a stale or small twist