The concept of who's "right" and who's "wrong" is just silly, especially when we're talking about two groups of people going to war with one another. It's natural for people to choose sides, that's what people do. When two of my mates are arguing, I normally agree with one over the other, just like when couples fight, people are bound to view one person in the relationship as being in the wrong and the other person being right. But that doesn't really translate to international relations where complex political problems don't always have short, simple answers.
It's not a case of the Americans being "right" and the British being "wrong" or vice versa. Each side could make a case for being right. It's all about one's own interpretation of the actual events.
After the signing of the Treaty of Paris, Great Britain agreed to formally recognise the United States as an independent sovereign nation. Because that occurred, we can't know what would have happened if the territory that's now known as the United States of America had remained British. Is it possible that the mass extermination of native Americans that occurred in the United States might never have happened? Sure it's possible, but for all we know it could have been a lot worse. Would things have played out in a similar fashion with slavery if the colonies had never achieved independence? Probably, because Britain abolished slavery before the United States did, but it's likely that the people who relied on slave labour and profited from the slave trade would have been just as upset if Britain had outlawed slavery as they were when the American government did. What they might have done about it is anybody's guess, but I wouldn't say that all out war would have been an option completely removed from the table. Would you?
And what about the way natives were treated in other places that remained under British rule? Some Australians treated Aborigines very harshly, and a few were especially cruel and brutal. And the Australian government - comprised of regents representing Great Britain, and later the UK, in Australia, failed to condemn them. In fact, it wasn't even illegal to kill an Aborigine in Australia until modern times as they weren't officially considered human beings. That's just despicable and inexcusable. Can you British claim some moral high ground that Americans cannot? I don't think so.
A good teacher presents facts and allows his or her students to form their own opinions. It's bad form to attempt to hammer one's own beliefs into a student and then cite examples and provide information that supports that view.
I'm not American and I'm not British, so I don't have a dog in the race on this one. But I can say that while the Americans were unspeakably immoral in their treatment of Native Americans, the British were no better when they were enslaving the Irish and sending whole families off to die on sugar plantations in the Caribbean, or when they were enslaving Africans, or when they were slaughtering people in Africa and Australia and India.
The answer to your question is that we all share the same history, but facts can be manipulated and the focus can be shifted. Things can be added, subtracted, altered, so the end result may be different. It's a sure sign of a weak mind for a person to be compelled to form an unwavering and rigid opinion on a broad and complicated topic. Intelligent people examine all the angles and see the situation as it is - with the good and the bad coming together to form a cohesive whole.