I currently live in the United States of America. I hope I will not offend anyone by abbreviating the country's name as "United States" or "U.S." I was born in the U.S., and I am a citizen of this country. I take no offense at being called an American. I take no offense at people in other countries in North and South America being called Americans, although I think it lends some ambiguity to the word "American".
When I refer to a date, I say "the third". If I need to specify a month to be clear, I say "March Third". If I need to specify the year as well, I say "March Third, Two Thousand Three". When speaking with people in other countries, I understand "The Third of March, Two Thousand Three" clearly, without confusion. I also imagine that I would understand "Two Thousand Three, March Three" or "March Two Thousand Three, Three" if anyone wanted to refer to the date that way.
I hope for everyone's sake that they can do the same. I thought this was a closed issue when it came to British and American spellings, and I thought this would apply as well. Use the date you are most comfortable with. Others will understand. If they don't understand, they should learn that the name of the month refers to the month, the big number refers to the year, and the small number refers to the day.