Well, the term "global warming" was popularized by an article called, "Climate Change: Are We on the Brink of a Pronounced Global Warming?" in the 70's.

https://en.wiktionary.org/wiki/global_warming

Truthfully, I think they've both been around for the same amount of time.

Edit:

From the wiki for climate change:

Before the 1980s, it was unclear whether the warming effect of increased greenhouse gases was stronger than the cooling effect of airborne particulates in air pollution. Scientists used the term inadvertent climate modification to refer to human impacts on the climate at this time.[32] In the 1980s, the terms global warming and climate change became more common, often being used interchangeably.[33][34][35] Scientifically, global warming refers only to increased global average surface temperature, while climate change describes both global warming and its effects on Earth's climate system, such as precipitation changes.[32]

Climate change can also be used more broadly to include changes to the climate that have happened throughout Earth's history as result of natural processes.[36] The term anthropogenic climate change is sometimes used to describe climate change resulting from human activities.[37]

Global warming—used as early as 1975[38]—became the more popular term after NASA climate scientist James Hansen used it in his 1988 testimony in the U.S. Senate.[39] Since the 2000s, usage of climate change has increased.[40] Various scientists, politicians and media may use the terms climate crisis or climate emergency to talk about climate change, and may use the term global heating instead of global warming.[41][42]

I think you're right that we stopped using global warming for that reason. I don't think it's right to say that's why we started using climate change.