Well, broadly speaking, there are two kinds of colonialism. Settler colonialism, and what I'll call 'exploitative' colonialism, which is kind of loaded, but it works for now.

The three countries you mentioned are all examples of settler colonialism. Large numbers of Europeans traveled to those countries and brought with them European institutions, religion and culture. You could argue that they were colonizers themselves, rather than colonized people, and that the truly colonized people were native Americans and Australian aboriginals, both of whom have quite a dark history of being nearly driven to extinction essentially. Of course, the line blurs a bit when you're talking about something like the American Revolution, where Americans expressed their view that Britain was a foreign colonizer in their lands, taking advantage of them through excessive taxation and so on. But in the earliest days of the American colonies, the distinction between Britain and the colonies was much more blurry. They were considered to be, and considered themselves to be, an extension of British civilization.

So rather than the people of the US, Canada and Australia being 'colonized' people, you can think of them as being collaborators in in a project to colonize other people, namely natives and aboriginals.

Now, take another colony, India. India was ruled by the British for hundreds of years, but there were always a relatively small number of actual British people living in India, unlike the US, Canada, and Australia, where Europeans make up the majority of the population today. And the fundamental purpose of this colonization was to make a profit. In the first phase of British rule in India, the country was actually governed by a private company called the East India Company, and their priority was to trade spices, which India had plenty of, and which of course fetched a high price in Europe. They also planted a lot of cash crops, such as opium, indigo, and cotton. But remember, the laborers on these farms were Indians. There were very few settlers in the American sense, farmers that made their homes in a new land.

India is a big place, and it took a while to develop a sense of national identity there, considering the number of languages and cultures and the history of it being fractured into many polities. But as that sense of national identity formed (some scholars would argue partially due to British colonialism itself), Indians began to agitate for independence from a nation that, while bringing infrastructure like the railway system and introducing the concept of democratic institutions, clearly prioritized profit over all else. And they won that fight, relatively bloodlessly, for reasons I won't go into here as they're not particularly important to answering your question.

So essentially, not all colonialism is the same. Some were projects of building European societies in new lands. Others were primarily business ventures. India in fact was somewhere in between. A better example would have been the Scramble for Africa, which I encourage someone else to go into.