Specific “Martini glasses” are cited by Barry Popik by 1934.
Martini’s are cited as a drink from about 1887.
MATINI & ROSSI AND ERNESTO ROSSI trademarked the term ‘MARTINI SOLA’ in 1916, as a descriptor for their Vermouth. More to the point, they first started using the term for commerce in 1882.
-no specific link because trademark search system links decay in a matter of minutes.
Says they, in their trademark application in 1916. Maybe they did, maybe they didn’t. Show me the proof.
I agree.
I know opinions will vary, but when I make mine at home I shake them, then pour them through a strainer only lightly held against the shaker; that engenders a fine mist of ice, such as you might find in the Arctic Ocean in late summer, that ensures sufficient coldness through the entire drink. There’s not much flavor at first, and then it opens up as I continue drinking it.
I believe I also read somewhere that a martini glass was supposed to be a shape which minimizes the area where glass touches liquid, to further minimize heat transfer. Note that normally a container designed for minimizing surface contact with its contents would be a sphere, but the problem changes considerably when one of the constraints is an open top. I’m not sure a cone is an ideal shape for this, but it seems like it might be, and if not, it’s close enough and stylish to boot.
But I hate martini glasses. The slosh factor is just far too high when the bulk of your drink is in the top half-inch of the glass. You might as well stick a saucer on a stick and pour my drink precariously into it. I make my martinis in highball glasses, make them super cold, and drink em quick. 
Well, you’re then left with what is essentially a small wineglass–which is shown in an illustration for the crusta in The Bar-Tender’s Guide by Jerry Thomas, which was the earliest known guide to making cocktails (from 1862). The crusta, incidentally, is not a cocktail in the traditional sense; it’s similar to a sidecar, which came several decades later.
The martini glass is essentially a modification of a standard piece of barware–small conical glasses were used for a long time prior to the martini, and the glasses we know as martini glasses are just a bit larger and relatively wider across the top.