I have been "warming the icecubes" for Pops ever since I was old enough to figure out 2/3 and 1/3 (wet gin martini...shudder). Taking a different angle, these are the things that make or break a 'tuni, a see-through, a silver bullet...for me. A generous and well made twist (use a peeler, and avoid pith at all costs), sufficiently chilled everything and shaken, cold liquor. Optimally the glass is filled to the absolute rim, pushing the limits of the meniscus surface tension. Shake until hands stick to the shaker, and when I look down at the resulting cocktail I want to be reminded of Shackleton.
Shaken martinis are also more effective in deactivating hydrogen peroxide than the stirred variety...essential obviously...further reading.
A point that is lost with many bartenders is the twist of the twist (more of a pinch really IMHO). The action allows a spritz (with a healthy plump lemon almost a viper spit) of lemon oil, that is visible on the surface (as below).
Zip the twist around the rim too, unless it is already submerged in vodka (as per meniscus comment).