Yes, although I am from Britain, I know that there are many different American accents and dialects. I can also just about tell the difference between American and Canadian and Australian and New Zealand, which many Brits can't do. I can only tell because of studying them.
American Canadian
"About" "Aboot"
Australian New Zealand
"Fish and chips" "Fush and chups"
One thing that annoys me, speaking of the "English accent" issue is people (including fellow Brits) saying that I am from England. As it happens, it is true that I am from England, I am ethnically English, but there are many people from the UK who are not.
Even though brothers and sisters who are Scottish, Welsh or Northern Irish are not nationalistic about their "country" they just find the geographical incorrectness of being called "English" annoying.
I usually refer to myself as being from Britain, or Great Britain, which by-the-way is the geographical name of the island I live on, it has nothing to do with the "greatness" of the nation, even though it is used that way a lot, especially by worldly patriots.
The Romans discovered that the natives of the island of the coast of Calais and the natives in north-west France were ethnically similar and spoke similar languages. They called the natives "Britons", and they called the island "Britannia", meaning "land of Britons". In the Middle Ages, Geoffrey of Monmouth in his work Historia Regnum Britanniae distinguished between the two areas inhabited by Britons by referring to the small part of France as "Britannia Minor" and the much larger island as "Britannia Major". Translated into English we get "Great Britain". Interestingly Ptolemy also used the term "Great Britain" to distinguish the large island from the smaller Ireland, which he called "Little Britain". North-west France is called "Brittany" and in modern French it is called "Bretagne", and the island is called "Grande Britagne". They are literally calling the island "Big Britanny".
So I tend to identify as being from the island Great Britain, but because of modern associations with the British Empire, I usually drop the "Great" among brothers in case they get the wrong impression.
Sorry for the long post, only geographical history is a passion of mine so inaccuracy is a pet peeve too.