Question: First let me assure everyone that I did look for a similar question and did not find any. So I apologize if my search was insufficient. 

I am a big fan of martinis, dry in particular. My buddies and I are partial to Leyden Dry Gin. If you have never tried it and have access to it, I highly recommend it. As a friend puts it, it provides better living through chemistry. It is from Holland, which, in case you did not know, is the birthplace of gin.

So my question, finally, is **at what point does a dry martini become simply a glass of cold gin?**

Answer: 
Dry Martini can sometimes become a very philisophical topic.

As Eric Shain already pointed out, the classic martini consists of gin and vermouth 6:1. You should not forget the water, that is melted while stirring it. This is quite an important part of making a good Martini. You should stirr it for several minutes.

More puristic people go to a ratio of 10:1 or even 15:1. To get to a more extreme combination, you put vermouth into your ice cubes, stirr it for a short while, and then pour the vermouth away, so that only the ice cubes for stirring your gin are thinly layered with vermouth. 

There are even recipes where you would simply put a bottle of gin in the freezer, put vermouth in an atomizer, then pour gin in a frozen martini glass, pfff vermouth once with the atomizer and voila - your martini. This is already very close to your suggestion of just drinking cold gin.

The famous Winston Churchill once said: "The best Martini is a bottle of gin, that stood in the shadow of a bottle of vermouth."

So there it is ;-)
I'm not an expert in this, but here is my take. A martini is a mixed drink. This suggests at least two ingredients. The typical dry martini is made from 5 parts gin and 1 part dry vermouth. Less dry vermouth is considered "drier". You can substitute vodka for gin. As far as I'm concerned, a dry martini becomes just gin when you stop including vermouth.