Yes, both terms existed since the 19th Century, but the concerted push on branding dates to the late 90s by green groups which began to discourage the phrase "global warming" and replace it with "climate change".
That's my recollection as well. There was a conservative meme in the late 90s/early 2000s about "Huh...it's cold during summer...how's that for global warming?", and I feel like "climate change" is a result of climate activist groups realizing global warming is a bad/confusing descriptor. I also recall in the early 2000s there was an attempt to get "global weirding" to replace "global warming" among activist communities, also seemingly in direct response to the conservative meme above.
That's correct. The idea that a Republican pollster like Frank Luntz is actually in control of climate messaging is not a serious claim, and I'm really confused why people are clinging to the idea that either the Republicans or "Oil Companies" are the ones responsible for the switch. That's bad history.