The Wikipedia article on Boston states that the first vowel in the name of the city is that of "caught," not "cot," citing Longman Pronunciation Dictionary. This seems consistent with my own experiences with chatting with Bostonians, who do not merge the two vowels. Many people around me do merge the two vowels, to whom this question is moot. How about other Americans who do distinguish "caught" and "cot," in, e.g., the Midwest and the South? Is the first vowel of "Boston" really the same as "caught" to them as well?
Your premise is wrong. In the traditional Boston accent, the vowels in cot and caught are merged, and are both pronounced with the caught vowel (although cart is pronounced the way the rest of the country pronounces cot).
Thus, Bostonians themselves generally pronounce the name of their city /bɔstən/, with the caught vowel, even those without the cot-caught merger.
If you are to believe the American Heritage dictionary, the rest of the country is confused as to how to pronounce it, and can use either pronunciation. I believe this is correct. But note that the rest of the country can't pronounce Worcester, Woburn, Haverhill, Quincy, Bowdoin, Cochituate and many other place names near Boston, either.