From the internet
Neither the term Dominion of Canada nor Dominion government appear in the 1867 Act; however, the former appears in the Constitution Act, 1871 — usage of which was "sanctioned" — and both appear in other texts of the period, as well as on numerous Canadian bills before 1967.
Until the 1950s, the term Dominion of Canada was commonly used to identify the country. As Canada acquired political authority and autonomy from the United Kingdom, the federal government began using simply Canada on state documents.
The transition away from the use of Dominion would later be formally reflected in 1982 with the passage of the Canada Act which referred only to Canada; later that year, the national holiday was renamed from Dominion Day to Canada Day. Section 4 of the 1867 BNA Act also declares that:
Unless it is otherwise expressed or implied, the Name Canada shall be taken to mean Canada as constituted under this Act.
and this has been interpreted to mean that the name of the country is simply Canada. No constitutional statute amends this name, and the subsequent Canada Act 1982 does not use the term dominion. However, the Canadian constitution includes the preceding BNA Acts, where the term is used; also, the Canada Act 1982 does not state that Canada is not a dominion. Official sources of the United Nations system, international organizations (such as the Organization of American States), the European Union, the United States, and other polities with which Canada has official relations as a state consistently use Canada as the only official name, state that Canada has no long-form name, or that the formal name is simply Canada. While no legal document ever says that the name of the country is anything other than Canada, Dominion and Dominion of Canada remain official titles of the country.
In recent years, the terms Dominion of Canada and Dominion are occasionally used to distinguish modern (post-1867) Canada from either the earlier Province of Canada or from the even earlier The Canadas. The terms are also used to distinguish the federal government from the provinces, though in this usage "federal" has become more common than "dominion". Among those who lament disuse of the term was the late Eugene Forsey, in response to what he and other monarchists consider increasing republicanism. However, the federal government continues to produce publications and educational materials that specify the currency of these official titles, although these publications are not themselves legal or official documents. For instance, in 2008 the Canadian government registered the Maple Leaf Tartan with the Scottish Tartan Authority under the name Dominion of Canada.
So, I guess we conclude with yet another typically Canadian solution: wishy-washy and middle of the road and inconclusive.