It's quite apt that if you search google for "Climate shift" it only gives you results for climate change. Which makes sense, because no one calls it "climate shift" as it's literally just a rewording for the same problem.
How I differentiate them:
Climate change = climate changing to a pattern completely at random.
Climate shift = climate patterns shifting / realigning with patterns of the past.
For examples sake, nobody was around say 5000 or 5 million years ago to know what the climate was.
Is it entirely possible that dinosaurs died off because of a massive heatwave that swept across continents?
Is it possible our climates are simply changing back to climate patterns of the past (not seen before by anyone alive/documented).