For those that wish to be traditionalists, a cocktail based on vodka and vermouth is called a Kangaroo. These days I am just happy when someone isn't referring to anything in a cocktail glass as a martini.
I mostly drink beer these days but I went through a cocktail phase. The way I like my martinis is pretty close to Eric Felten's "Original Intent Martini" and I tend to prefer Plymouth and Noilly Prat. I also occasionally like a perfect martini, perfect in this context meaning equal parts of dry and sweet vermouth.
I stir but I think the distinction is overblown. There is a subtle difference in texture and an obvious difference in clarity, but other than that I would spend more time convincing people to use more vermouth than convincing them to stir.
James Bond had an excellent reason for asking for his drinks shaken. He was worried that someone would dip the stirring stick in poison. I don't know why they couldn't have put poising in the shaker or the glass, but I didn't write the books. I will say if you have never made a Vesper, it's not a bad cocktail.