What was America called under British rule?

The question will be taken to refer to what America was called in the time leading up to the American Revolution, in the decades just prior to independence.
The most direct way of answering this question is to go straight to the primary sources. In repealing the Stamp Act in 1766, the British Parliament wrote of "the British colonies and plantations in America." In the Albany Plan of Union in 1754, Benjamin Franklin and Thomas Hutchinson, aside from mentioning the colonies by name (e.g. Massachusetts Bay and New Hampshire), refer to the "colonies" collectively and to "America," seemingly having in mind the collective territory of the colonies. If we look at other documents of the era, we will see "America" and "North America" used somewhat interchangeably. We will also see references to certain regions, like "New England."
You can probe this question further by examining eighteenth-century primary sources at the third link provided below. If you're interested in the evolution of the names and labels over time, the Avalon Project also has documents from the preceding two centuries, as the English colonized and settled in North America.
https://avalon.law.yale.edu/18th_century/albany.asp

https://avalon.law.yale.edu/18th_century/repeal_stamp_act_1766.asp

https://avalon.law.yale.edu/subject_menus/18th.asp