Some of the confusion about the name Canada is caused by 'when'. The country of Canada did not exist until 1867. Prior to 1867 the name was commonly used to include large parts of North America, many of these areas are not now part of the country of Canada. If we are talking about pre 1867 we should be careful to use the name that was in use at the time. I don't know how most residents of the area referred to the larger area at that time. I do know that most of my husband's ancestors list their birthplace as Canada on the 1851 census.
In fact, when my husband's ancestor emigrated to what is now upper New York State in 1768 he was actually moving to the Province of Quebec. He probably thought he was going to 'the colonies'.
The province of Quebec, name use from 1763-1776 included many parts of the now northern United States including New Hampshire, Massachusetts, Connecticut, New York, New Jersey, Pennsylvania and Maryland.
The name “Canada” likely comes from the Huron-Iroquois word “Kanata,” meaning “village” or “settlement.” In 1535, two Aboriginal youths told French explorer Jacques Cartier about the route to Kanata; they were actually referring to the village of Stadacona, the site of the present-day City of Québec. For lack of another name, Cartier used the word “Canada” to describe not only the village, but the entire area controlled by its chief, Donnacona.
The name was soon applied to a much larger area; maps in 1547 designated everything north of the St. Lawrence River as Canada. Cartier also called the St. Lawrence River the “rivière du Canada,” a name used until the early 1600s. By 1616, although the entire region was known as New France, the area along the great river of Canada and the Gulf of St. Lawrence was still called Canada.
Soon explorers and fur traders opened up territory to the west and to the south, and the area known as Canada grew. In the early 1700s, the name referred to all French lands in what is now the American Midwest and as far south as present-day Louisiana.
The first use of Canada as an official name came in 1791, when the Province of Quebec was divided into the colonies of Upper Canada and Lower Canada. In 1841, the two colonies were united under one name, the Province of Canada, from 1841 to 1867.
The British settlers had originally called the land that covered Quebec, Ontario, and part of the United States all as Quebec. It wasn’t until the British enacted the Constitutional Act in 1791 that Ontario would be known as the land upstream from the St. Lawrence River, or Upper Canada, and Quebec considered the land downstream from the St. Lawrence River, known as Lower Canada. In 1867, Ontario and Quebec officially became separate provinces.
One term commonly used by Family Search that I would like to see disappear is British North America, it's use seems to have originated from the name of the act that created the country of Canada. It was called The British North America Act but it did not mean that was what the area was called except perhaps by bureaucrats.