I thought that when I moved to the UK, I'd no longer have to worry about specifying. I was wrong. It's even worse. The well-educated will still ask me whether I mean the York-in-Canada or the one in the UK, especially if they know I just moved here from Canada. A large number of others will mishear me as saying my MA is from New York (It's my funny foreign accent), and then either ask which institution (I just told them!) or just cruise along under the wrong assumption.
There's a London ad campaign which has similarly caused me confusion. When I was moving to London, most people I knew in Ontario needed to know which London - London, Ontario or London, England? When I arrived in London, large posters on the Underground proudly proclaimed "EVERYONE'S LONDON". The emphasis on the final ON made me think "Ontario" every single time. Why was the ad campaign emphasizing the "on"? I have no idea. It gave the advertising all the wrong spin to my biased brain.
There are too few names in the world. I went to a book launch for a book named Nova Scotia - it has nothing to do with Canada, and everything to do with "New Scottish Stuff". And so it goes. I'll continue to need to specify which York I went to - and which of three countries it's located in.