"Responsible government" means the country governed themselves. They were the dominions of white settlers. Because they were European in race, culture, and experience with self-rule, they were allowed to govern themselves. This was to prevent an American Revolutionary War scenario where colonists might attempt a war for independence if they were frustrated in their attempts to govern themselves.
Initially, the dominions under responsible government were still dependent and influenced by the British parliament, but they were given equal status under the Statue of Westminster in 1931.
In areas where white settlers were not dominant enough, the British exercised direct rule through a governor general and reported to London.
India was a special case. It was allowed some measure of self government, but not fully. The Indian Congress was always agitating for more self-government which the British were reluctant to provide for various reasons. One was that although India had a large body of educated elites, the vast majority of its people remained ignorant and poor. Plus, to be perfectly honest, Indians weren't white, and the British wanted to preserve the privileges of British settlers there. Plus the Indian Muslims were hesitant about being in a Hindu dominated democracy which would place Muslims under Hindu rule. And India itself was divided into areas ruled directly by the Crown (through a governor general) and the Princely states (which were ruled by their princes under suzerainty of the British Crown - this would be considered indirect rule.) Because there was such a wide variety of arrangements, fashioning a common Indian government was hard. The British usually recommended dividing up India into more easily governed states which could govern themselves and keeping the princely states separate from other areas, but this was rejected by Congress which did not want India divided.
And of course, there were protectorates effectively ruled by the British, but supposedly independent, like Egypt.
So yes, British governance of its empire constantly changed based on the needs of the empire and the local population. As events changed, so did the status of the colony. Usually this meant more self-rule, but not always. Newfoundland was originally not part of Canada, but its own dominion under responsible government. During the Great Depression, the government couldn't meet its financial obligations, and London took over. Eventually Newfoundland was allowed to decide its own fate and joined Canada instead.